FlowMixer: A Depth-Agnostic Neural Architecture for Interpretable Spatiotemporal Forecasting
Authors: Fares B. Mehouachi
Saif Jabari
Venue: NeurIPS 2025 poster ()
Keywords: forecasting
spatiotemporal forecasting
neural architecture
interpretability
Koopman operators
dynamical systems
long-horizon prediction
non-negative matrix factorization
Primary area: machine_learning_for_sciences
TL;DR: We propose FlowMixer, a single-layer architecture using semi-group properties to eliminate neural depth search, achieving competitive multivariate forecasting with interpretable Kronecker-Koopman eigenmodes and algebraic horizon manipulation.

Abstract: We introduce FlowMixer, a single-layer neural architecture that leverages constrained matrix operations to model structured spatiotemporal patterns with enhanced interpretability. FlowMixer incorporates non-negative matrix mixing layers within a reversible mapping framework—applying transforms before mixing and their inverses afterward. This shape-preserving design enables a Kronecker-Koopman eigenmodes framework that bridges statistical learning with dynamical systems theory, providing interpretable spatiotemporal patterns and facilitating direct algebraic manipulation of prediction horizons without retraining. The architecture's semi-group property enables this single layer to mathematically represent any depth through composition, eliminating depth search entirely.
Extensive experiments across diverse domains demonstrate FlowMixer's long-horizon forecasting capabilities while effectively modeling physical phenomena such as chaotic attractors and turbulent flows. Our results achieve performance matching state-of-the-art methods while offering superior interpretability through directly extractable eigenmodes.
This work suggests that architectural constraints can simultaneously maintain competitive performance and enhance mathematical interpretability in neural forecasting systems.

FlowMixer: A Depth-Agnostic Neural Architecture for Interpretable Spatiotemporal Forecasting
Fares B. Mehouachi
Saif Eddin Jabari New York University Abu Dhabi, UAE & Brooklyn, USA sej7 rej7@nyu.edu
New York University in Abu Dhabi Abu Dhabi, UAE
fm26206nyu. edu
Abstract
We introduce FlowMixer, a single-layer neural architecture that leverages con- strained matrix operations to model structured spatiotemporal patterns with en- hanced interpretability. FlowMixer incorporates non-negative matrix mixing lay- ers within reversible mapping framework-applying transforms before mixing and their inverses afterward. This thape-preserving design enables a Kronecker- Koopman eigenmodes framework that bridges statistical learning with dynamical systems theory, providing interpretable spatiotemporal patterns and facilitating direct algebraic manipulation of prediction horizons without retraining. The archi- tecture's 's semi-group property enables this single layer to mathematically represent any depth through composition, eliminating depth search entirely. Extensive experiments across diverse domains demonstrate FlowMixer's long-horizon fore- casting capabilities while effectively modeling physical phenomena such as chaotic attractors and turbulent flows. Our results achieve performance matching state-of- the-art methods while offering superior interpretability through directly extractable eigenmodes. This work suggests that architectural constraints simultaneously maintain competitive performance and enhance mathematical interpretability neural forecasting systems.
Introduction
The ability predict complex spatiotemporal patterns remains fundamental challenge across scientific disciplines, including climate modeling, biological systems, and fluid dynamics [1, 2, 3]. Statistical methods have traditionally dominated time series forecasting [4], while mechanistic models form the backbone of dynamical systems analysis [5]. Recent advances Machine Learning have demonstrated that these approaches be complemented and extended through data-driven methods [6, 7]. For example, reservoir computing, originally designed for chaos prediction [1,7, 7, 8). has shown promise in statistical forecasting [9, 10], while neural architectures for time series forecasting have successfully captured chaotic dynamics [1]]. This convergence hints at the possibility of a unified framework for spatiotemporal pattern learning [12, 13].
Achieving this integration presents several challenges. Time series data often exhibit statttatiin,r,ty, where statistical properties evolve over time, creating distribution shifts between training and testing [14]. Dynamical systems, although low dimeemmssional, may exhibit chaotic behavior characterized by extreme sensitivity initial conditions [15, 16, 17, 18). Additionally, the differing operational requirements of these tasks ttsks-typ multi-step predictions for time series and iterative one-step predictions for dynamical systems demand flexible and efficient temporal modeling capabilities.
Recent neural architectures have made significant progress addressing these challenges. Transformer-based models like PatchTST [19] effectively capture long-range dependencies through attention mechanisms and patch tokenization. MLP-based architectures such as TimeMixer+ [20]
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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and WPMixer [21] achieve impressive efficiency through multi-scale decomposition and structured mixing operations. State space models, including Chimera [22], offer promising capabilities for modeling multivariate time series with coupled dynamics. Meanwhile, Gilpin [23] has shown that large-scale statistical models can effectively capture the behavior of chaotic systems, while foundation models [24, 25] demonstrate emerging capabilities for few-shot and zero-shot forecasting. These architectures typically require extensive hyperparameter search to determine optimal network depth, which varies significantly across datasets and domains.
Kronecker Kookken Eigenmodes
Extended Forecast
Input Sequence
Forecast
Unterpreeattoo))
Features
Time
Wy & = PDP-1
DMD Diagonalization
Y x- (2)
Mixing Layer
X
W. a W W wW,
ReviN
(X)
W)
Z
ReviN denorm
Man (1+99)
W
Time Mixing
Features Mixing
Figure 1: Overview of FlowMixer's single-layer constrained architecture. The core architecture (bottom) processes input sequences using three key components: (1) reversible mapping, mainly reversible instance normalization (RevIN) to handle distribution shifts, (2) a constrained mixing layer with non-negative time mixing (W;) and stochastic feature mixing (W,) matrices, and (3) adaptive skip connections embedded within the mixing matrices. The spatiotemporal evolution (top) demonstrates how data padded to create square mixing matrices, enabling eigendecomposition and efficient temporal modeling and interpretation. The Kronecker-Koopmal eigenmodes (right), derived from the architecture's 's mathematical structure (W, @ W, = PDP-1), provide a space-time decomposition of the mixer input weighted sum of space-time eigenmodes.
Here, we introduce FlowMixer (Fig. 1), a neural architecture designed to address these challenges through constrained design. Inspired by non- non-negative matrix factorization [26] and Dynamic Mode Decomposition (DMD) [27]. FlowMixer aims to draw connections between statistical learning approaches and dynamical systems theory while providing mathematically interpretable framework. FlowMixer explores a different point in the design space: rather than optimizing architectures for individual domains, demonstrate that careful constraints enable single architecture to achieve competitive performance across time series forecasting, with additional capabilities in chaos prediction and turbulent flow modeling. This cross-domain versatility, combined with interpretable eigenmodes and elimination of depth search, offers practical advantages for applications requiring consistent behavior across varied spatiotemporal phenomena.
Our contributions include:
constrained architecture whose semi-group property eliminates depth as hyperparameter: single layer is all you need.
A shape- preserving design that enables direct extraction of Kronecker Kronecker-Koopman eigenmodes for interpretability and allows algebraic prediction horizon change without retraining.
A simplified chaos prediction approach through Semi-Orthe Semi-Orthogonal Basic Reservoir (SOBR)
that matches specialized frameworks while maintaining architectural simplicity.
Empirical validation across time series forecasting, chaotic systems, and turbulent flows demonstrating competitive cross-domain performance.

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2 Related Work
2.1 Time Series Forecasting Architectures
Recent advances time-series forecasting have focused on improving accuracy through architectural sophistication. TSMixer [28] employs alI-MLP time and feature mixing but requires empirical deter- mination of optimal layer count. TixeMixer++ [20] employs hierarchical multi-scale decomposition with bidirectional mixing for seasonal and trend patterns. Chimera [22] extends state space models to multivariate time series through a 2D SSM architecture with three-headed processing along time and variate dimensions. While these methods achieve strong performance, they compound the deployment challenge: each new dataset potentially requires extensive architecture search from scratch.
Koopman-Based Neural Methods
The integration of Koopman operator theory with neural networks has produced sophisticated forecasting frameworks [29, 30, 31]. Koopa [29] employs multi-layer architectures with eigenvalue- based stability checking and explosion prevention mechanisms, while KNF [30] utilizes three-term loss function across various network components. Both approaches treat the Koopman semi-group property K, o K, Kt++ embedded within neural network innccessiaeee for direct analysis.
optimization target. Notably, their eigenmodes remain deeply
2.3 Cross-Domain Spatiotemporal Methods
Few architectures attempt unified modeling across statistical and physical domains. Reservoir computing [1, 7] bridges chaos prediction and time series but requires careful tuning of reservoir size, spectral radius, and input scaling. N-BEATS [32], though designed for time series, demonstrates exceptional performance across chaotic systems [11], suggesting untapped cross-domain potential.
FlowMixer sits at the intersection of these research directions: it adopts the mixing paradigm from time series architectures, guarantees Koopman's 's mathematical properties by construction, and achieves cross-domain applicability through a single constrained design.
Flowr Architecture
FlowMixer builds upon TSMixer [28] by addressing the challenge of layer count optimization. Instead of empirical layer stacking, we developed a single mixing block with semi-group properties where successive applications combine algebraically (see Section 4. This approach eliminates depth as hyperparameter while enabling mathematical manipulation of prediction horizons (see Section 4.3). The various components of this architecture are presented hereafter.
3.1 Formal Definition
Let X Reexny be input tensor, where n, denotes the number of time steps and ny the number of features. FlowMixer defines a transformation F m xn, + Ruxx as:
F(X, W., Wr Φ) = o ' (W,O(X)W?)
(I)
where á R + 1. is reversible mapping. W, R" is a time mixing matrix, and W, X is feature mixing matrix. FlowMixer preserves tensor dimensions: the output F(X) Raxxa approximates a target Y of the same shape, where the last h rows of Y contain the forecast and earlier rows contain historical data. Padding with historical data, ensures square mixing matrices, enabling eigendecomposition and semi-group composition. The name "FlowMixer" reflects this reversible flow of data around the central mixing operations.
3.2 Core Components
Dependeptn Reversible Instance Normalization: We employ mainly RevIN [33] as ø for feature-wiss normalization, extending it to Time- -emenDependent RevIN (TD-RevIN):
RevIN(ax) Zt - -Err]]
E[xe]
TD-RevIN(<:)) = at /Va((x)) +
br (2)
/Vr((z)) +c

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where de, by are time-varying parameters, and x denotes the Hadamard product. This allows different normalization parameters for each timestep, enhancing the model's ability to handle non-stationary patterns. This TD extension is possible thanks to the preppeppeeeeving property of FlowMixer.
The core functionality integrates principles from non-negative matrix fac- torization and attention mechanisms [26, 34]. For an input tensor X, our architecture applies two complementary transformations:
Constrained Mixing:
TimeMix(X)= = W,X, (x)) = XW]
The time mixing matrix Wt aligns with Koopman theory through scaled matrix exponential:
W, ae (WoXW)) Rs a/ + aWo x Wo
where denotes Hadamard (element-wise) product. a is scaling learnable parameter initialized at one that controls memory persistence. In practice, we use this first-order approximation with learnable matrix W that absorbs the scaling, yielding W; = al + W x W. This design creates an analog to fractional differencing (FARIMA) [35] that adapts to varying temporal dependencies (see Appendix J). The nonnegative quadratic term forces the model to capture temporal dependen- cies through positive interactions only, reducing spurious correlations while introducing nonlinear interactions.
For feature mixing, we implement a static attention mechanism:
I I+ 3 SoftMax(QK" /vda)
w
(5)
1+ 32
where K,Q E R are learnable key/query matrices, d, is the multiplication dimension of K, Q, and modulates attention strength. This formulation has graph-theoretical interpretation as transition probabilities a weighted directed graph, analogous to Markov process.
Scasonalities: For time series with inherent periodicities, we enhance the time mixing operation by leveraging Kronecker structure:
W. = Σ WrOn & WT, x (p) = n4+
where W, E RPXP captures patterns at the period level Wr(p) E R models relationships between different period blocks, and p = ny. This formulation enables efficient modeling of multiple seasonal patterns while preserving the simplicity of the architecture.
SOBR: Ortioortl Basic Reservoir. For chaotic systems, introduce SOBR-a simplified reservoir computing approach that uses random, non trainabee semi-orthogonal matrices. SOBR replaces á with S where S (X) (U,)U))) and a is an invertible activation (e.g., Leaky ReLU). These matrices are constructed to satisfy:
U,UT = Ta,, U.T: = la.
This expansion provides a digees-ai representation space appropriate for modeling chaotic systems (discussed in Section 4).
Theoretical Analysis
The architectural constraints of Flow yield unique theoretical properties that bridge concepts from statistical learning with dynamical systems theory [5, 36). The key insights emerge from the architecture's semi-group stability.
4.1 Semi-Group Structure
Square mixing matrices and invertible standardization enable semi-group Sabbiltty-composiite of Flow Mixers with identical standardization Φ remain a FlowMixer:
F(F 0' Φ) F(z, 0".
(8)

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This property fundamentally distinguishes FlowMixer from all existing forecasting architectures: while traditional architectures require depth optimization and alter capacity with each layer, any n-layer FlowMixer composition has an equivalent single-layer representation'. This mathematical guarantee eliminates architecture search entirely and enables flexible horizon adjustments without retraining.
4.2
Eigen Decomposition and Kronecker- Kooppan Framework
In vectorized form, FlowMixer's mixing operation reveals Kronecker product:
vec(W;XW7) = (Wy X We)vee(X)
(9)
This enables the extension of classical Dynamic Mode Decomposition (DMD) [27] coupled spatiotemporal patterns in novel principled way without Hankelization [37, 38J. Given eigendecom- positions Wy PDP-1 and We = QEQ- where = diag(A,) and E' diag(u;), any input matrix X can be expressed as:
x Σ (an((1)))) 1.
(10)
where Py are columns of Q and P, and ai, = "P-' are projection coefficients.
Unlike Koopman neural methods where eigenmodes remain embedded [29, 30], these modes directly extractable through standard matrix decomposition. Both time and feature eigenvectors form bases in their respective spaces. These components form the Kronecker-Koopne (KK) spatiotemporal eigenmodes:
i. qi 8 Pj. (1,j) E [1, nc] x [1,
(11)
The Kronecker product preserves completeness, yielding n, x ny basis eigenmodes spanning all possible spatiotemporal patterns. These modes analogous to Koopman modes [39, 40]. but naturally couple spatial and temporal patterns. Section 5.2 provides an example visualization of these
eigenmodes.
4.3 Continuous Algebraic Horizon Modification
The KK structure enables direct algebraic modification of prediction horizons. For horizon change from ho to h and a ratio & = hi/ho, we can express the modified prediction as:
(t) Σ oi
(pP)
exp
tlog
(Nafty)
(12)
KK Eigenmodes
KK Figenvalues
This enables continuous horizon adjustment through interpolation < 1) or extrapolation (See Appendix H.2). The framework extends naturally to derivatives. For >> 0:
x
aij (g:P)) (Aa11)) (log (AaHz))
(13)
0ta
2
The eigenmodes structure remains invariant during these modifications, with temporal evolution controlled mainly through eigenvalue scaling. Stability requires that all eigenvalues are inside the unit circle and can be implemented through unitary constraints on time mixing matrices [41]. The feature matrix is already stochastic and verifies this property.
Experimental Validation
5.1 Long-Horizon Time-Series Forecasting
Time series forecasting presents persistent challenges, particularly for extended prediction horizons where emoor accumulation compounds modeling difficulties. We evaluate FlowMixer benchmark
For SOBR models, semi-group property holds in lifted space.

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datasets widely used in the forecasting literature, including ETT (electricity transformer temperature), Weather, Electricity, and Traffic datasets [42]. Our comparisons include recent architectures such as TimeMixer++ [20], Chimera [22], MSHyper [43, 44]. CycleNet [45], and TSMixer [28]. Flow Mixer deliherately forgoes the architectural complexity of these methods: no hierarchical decomposition, no multi-scale processing, no specialized seasonal modules. While the semi-group property guarantees composition e.g. F(96) o F(96) = F(192), enabling algebraic horizon modification, we train each horizon independently for fair comparison (see Appendix H). The practical implication: practitioners can directly transfer FlowMixer between projects without architectural modifications. Implementation details and hyperparameters are provided in Appendices A and C, respectively.
Table 1: Performance comparison of time series forecasting models on multiple datasets across different prediction horizons. Results show Mean Squared Error (MSE) and Mean Absolute Error (MAE), with lower values indicating better performance. Bold blue is best, orange is second best. FlowMixer achieves comparable performance using a single operational layer, eliminating the depth hyperparameter search required by all baseline methods. Statistical validation via Wilcoxon signed-rank tests confirms significance (p<0.05, see Appendix D).
FlowMixer
Chimera
TimeMixer++
iTransformer (2024)
Ada-MSHyypr (2024)
CycleNet (2024)
TSMixer (2023)
(Ours)
(2024)
(2024)
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAB
MSE
MAE
MSE
MAE
MSE
MAB
96
0.355
0.387
0,366
0.392
0,361
0.403
0,386
0.405
0.372
0.393
0.375
0.395
0,361
0.392
192
0.394
0.409
0.402
0.414
0.416
0,441
0,441
0,436
0,433
0.417
0.436
0.428
0,404
0,418
336
0.415
0.423
0.406
0.419
0.430
0.434
0.487
0.458
0.422
0.433
0.496
0.455
0.420
0.431
720
0.433
0.454
0.458
0.477
0.467
0.451
0.503
0.491
0.445
0,459
0.520
0.484
0.463
0.472
96
0.264
0.330
0.262
0.327
0.276
0.328
0.297
0.349
0.283
0.332
0.298
0.344
0.274
0.341
192
0.320
0.368
0.320
0.372
0.342
0.379
0,380
0.400
0.358
0.374
0.372
0.396
0.339
0.385
336
0.344
0.396
0.316
0.381
0.346
0.398
0.428
0.432
0.428
0.437
0.431
0.439
0,361
0.406
720
0.388
0.452
0.389
0.430
0.392
0,415
0.427
0.445
0,413
0.432
0.450
0.458
0.445
0.470
96
0.298
0.345
0,318
0.354
0.310
0.334
0.334
0.368
0.301
0.354
0.319
0.360
0.285 0.339
ETTm2 ETTml
192
0.333
0.365
0.331
0.369
0.348 0.362
0.377 0.391
0.345
0.375
0.360
0.381
0.327
0.365
336
0.360
0.386
0.363
0.389
0.376
0.391
0.426
0.420
0.375
0.397
0.389
0.403
0.356
0.382
720
0.398
0.410
0.409
0.415
0.440
0.423
0.491
0.459
0.437
0.435
0.447
0.441
0.419
0.414
0.159
0.252
0.169
0.265
0.170
0.245
0.180
0.264
0.165
0,257
0.163
0.249
0.163
0.252
192
0.211
0,290
0.221
0,290
0.229 0,291
0,250 0,309
0,230
0.307
0.229
0.290
0,216
0.290
336
0.260
0.324
0.279
0.339
0,303
0,343
0,311
0.348
0.282
0.328
0.284
0.327
0,268
0.324
720
0.333
0.377
0.341
0.376
0.373
0.399
0.412
0.407
0.375
0.396
0.389
0.391
0.420
0.422
96
0.143
0.194
0.146
0.206
0.155
0.205
0.174
0.214
0.157
0.195
0.158
0.203
0.145
0.198
Traffic Electricity Weather
192
0.185
0.235
0.189
0.239
0.201 0.245 0,221
0.254
0.218
0.259
0.207
0.247
0.191
0.242
336
0.235
0.276
0.244
0.281
0.237
0.265
0.278
0.296
0.251
0.252
0.262
0.289
0.242
0.280
720
0.305
0.326
0.297
0,309
0.312
0.334
0.358
0.347
0.304
0.328
0.344
0.344
0,320
0.326
0.131
0.226
0.132
0.234
0.135
0.222
0.148
0,240
0.135
0,238
0.136
0.229
0.131 0.229
192
0.146
0.239
0.144
0.223
0.147
0,235
0.162 0,253 0.152
0.239
0.152
0.244
0.151 0.246
336
0.160
0.255
0.156
0.259
0.164
0.245
0.178
0.269
0.168
0.266
0.170
0.264
0.161
0.261
720
0.195
0.289
0.184
0.280
0.212
0.310
0.225
0.317
0.212
0.293
0.212
0.299
0.197
0.293
96
0.377
0.264
0.366
0.248
0.392
0.253
0.395
0.268
0.384
0.248
0.458
0.296
0.376 0.264
192
0.388
0.268
0.394
0.292
0.402
0.258
0.417 0.276
0.401
0.258
0.457
0.294
0.397 0.277
336
0.401
0.275
0.409
0.311
0.428
0.263
0.433
0.283
0.423
0.261
0.470
0.299
0.413 0.290
720
0.434
0.291 0.443
0.294
0.441
0.282
0.467 0.302 0.453
0.282
0.502
0,314 0.444 0.306
Table demonstrates FlowMixxrr' competitive performance across multiple datasets and horizons, with particularly strong results on ETTm2 and the 720-timestep horizon for Traffic and ETT fore- casting tasks. What distinguishes FlowMixer is its architectural simplicity-unlike contemporary models with deep architectures and hierarchical decompositions, it achieves these results consistently using just a single operational layer with carefully designed constraints. The computational efficiency directly benefits from this simplicity, with FlowMixer processing most ETT datasets in under 2 sec- onds per epoch (see Appendix C). Statistical validation through Wilcoxon signed-rank tests confirms that these improvements are significant (p<0.05, see Appendix D). Ablation studies confirm these architectural choices contribute to the model's 's forecasting capabilities (Appendix F), demonstrating how carcfully targeted constraints enhance multivariate time series prediction.

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5.2 Kronecker-Koopman Analysis of Spatiotemporal Patterns
FlowMixer offers a novel perspective on spatiotemporal analysis beyond the traditional Dynamic Mode Decomposition (DMD) framework [27]. Combining time and feature spectral elements yields the Kronecker- Kronecker-Koopman (KK) eigenmodes (see Section 4), To illustrate these patterns, we compute KK eigenmodes from the traffic dataset [46] enabling effective visualization of spatial components.
a
elgenvalues
eigenvector de
eigenvector
eigenvector US
0.10
0.20
Imaginary
0:13
Value
0
0:10
0.05
Time (Days)
Time (Days)
(Days)
Feature eigenvector V:
Ste(Vo @ ugst
ReVVV u
3ee(V: @ uz}
x30
K307
0.033
0.035
Volue
Feature elgenvector V
Re(V's @ v1))
Ree(Vss u]
Mee(Va @ (g)
x30-2
200
830
400
0.0
600
800
1.5
800
0608
3.00
0.26
Value
Time (Days)
Time (Days)
Time (Days)
Figure 2: Visualization of Kronecker- Kronecker-Koopmag Koopman Eigenmodes for traffic dataset. (a) Time eigenvalues distribution, with aligned angular values indicating the periodic nature of the traffic datasets. (b-d) First three time eigenvectors (real part). (e,i) First and Third space eigenvectors, with the first showing constant values consistent with the stochastic space mixing matrix (representing Markov transition process). (f-h, j-l) Real parts of Kronecker-Koopman Eigenmodes revealing space-time patterns. The periodic structures in time eigenvectors and coherent patterns higher-mode products demonstrate how FlowMixer captures spatiotemporal dynamics.
When applied traffic flow data, KK analysis yields structural insights. The time eigenvalues distribu- tion (Figure 2a) reveals clear angular alignments corresponding to daily, weekly, and intra-day traffic periodicities. The primary time eigenvector (b) captures baseline (trend) traffic volume with weekend transitions, while subsequent eigenvectors (c-d) exhibit accelerating oscillatory behavior (seasonali- ties) reflecting 24-hour and 12-hour cycles. Feature eigenvectors (e-g) encode spatial relationships between traffic sensors from global to local scales, with the dominant eigenvector exhibiting the ex- pected constant structure characteristic left stochastic matrices. The resulting Kronecker Koopmann spectrum, especially higher-order modes (j-l), manifests coherent cross- dimenssina patterns that reflect propagation phenomena consistent with congestion dynamics traffic theory [47, 48].
This framework provides spatiotemporal modes beyond classical DMD [27] and Hankelized- DMD [37, 38] by separating temporal and spatial components instead of relying on Hankelized embeddings. Our approach captures the inherent structure of any system where spatiotempo- ral patterns govern dynamics, and expands the traditional time series dichotomy, from a classic trend/seasonality decomposition to spatiotemporal framework:
+ 3 :local) T
01
1/
ttrend eseasonal
1/1
KK treend
sccaoomal
4
glohal
(14)
7

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5.3 Prediction of Chaotic Dynamical Systems
The prediction of chaotic systems represents of science's fundamental challenges, where expo- nential sensitivity to initial conditions traditionally limits long-term forecasting [18]. Specialized
architectures like reservoir computing have advanced this field significantly [1, 12, 49, 7, 50], often focusing on recurrent frameworks for iterative single-step predictions. We explore how FlowMixer's constrained design principles can contribute this domain through complementary approach.
The key component is SOBR (Semi-Orthogonal Basic Reservoir)-an adaptation of reservoir com- puting principles that aligns with FlowMixer's constrained design philosophy (cf. Section 3.2). While Machine Learning approaches have traditionally addressed nigh-dimensional problems [51], chaotic systems present different challenge: complex dynamics emerging from low dimensions (e.g., the three-dimensional Lorenz. system). SOBR addresses this dimensional consideration through targeted expansion: projects data into nigher-dimensional spaces using random semi-orthogonal matrices, maintaining full rank and unit spectral radius. This Koopman-inspired dimensional lifting enhances FlowMixer's pattern recognition capabilities. FlowMixer's results for chaotic attractors without SOBR are provided in Appendix K.3.
SOBR integrates with Flowrixer through the reversible mapping é. In its simplest form it is expressed as (X) = (U,,) where cemi-orthogonal matrices Uy and Ug expand feature and time dimensions respectively, complemented with an invertible activation a (typically Leaky ReL.U). This design enhances FlowMixer's ability to capture chaotic attractor dynamics. While this SOBR integration preserves the gooup property only in the lifted space, not the original space (see Appendix K.2), the benefits of dimensional lifting outweigh this tradeoff for chaotic systems, where accurate pattern recognition takes precedence over direct algebraic horizon manipulation.
Lorenz
b
Rossler
FlawMiixrr
Divergence
Rl: converpence
200
400
600
800
1000
200
430
800
1000
200
400
600
800
1000
Value
200
400
600
800
1000
200
4.00
600
800
1000
200
400
600
800
1000
Value
Reservoir
Divergence
Reservplr Lag
Time Step
Time Step
Time Step
Ground Truth
NBEATS
Reservoir
FlowMixer
Figure 3: Predictions of Lorenz (a), Rössler (b), and Aizawa (c) chaotic attractors (scaled [-1,1]. Each row shows the evolution of x, and z variables over time. Ground truth (black), FlowMixer (blue), Reservoir Computing (orange) [1, 52, 53), and N-BEATS (purple) [32] trajectories While all methods capture the Lorenz attractor's structure, N-BEATS shows a notable deviation in the Rössler system, and Reservoir Computing shows differences particularly in the z-component predictions. FlowMixer exhibits consistent performance across all three systems. Experimental settings are summarized Appendix C and presented in detail in Appendix K.4.
compared.

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We evaluate this approach on three canonical chaotic systems: the Lorenz butterfy [15], Rössler attractor [16], and Aizawa system [17]. Rather than employing iterative single-step predictions, FlowMixer processes 32-step sequences to forecast the next 32 steps directly. By compounding these predictions, we obtain forecasts over 1024 steps aossapppppiiaalllyyy 9-10 Lyapunov times for the Lorenz system. Figure 3 compares FlowMixer's performance with Reservoir Computing (RC) [1, 52, 53] and N-BEATS [11, 32j. While all methods capture the general attractor structure, they exhibit different failure modes: RC struggles with accurate z-component predictions particularly in the Lorenz system, N-BEATS diverges notably in the Rössler system, while FlowMixer maintains consistent accuracy across all three attractors. The model captures characteristic features from the Lorenz butterfly to the Aizawa spiral dynamics, occasionally realigning with ground truth trajectories effective pattern learning. These results illustrate how architectural constraints can maintain predictive capabilities for chaotic systems. FlowMixer with SOBR offers an alternative approach that complements existing specialized frameworks. This same single-layer architecture that forecasts time series also captures chaotic dynamics. This connection between statistical and dynamical modeling suggests broader implications for scientific computing, which we explore next through turbulent flow prediction.
after Seerrgnnee-uugggeeee
5.4 Predicting 2D Turbulent Flows
The prediction of turbulent flows represents a crucial challenge at the intersection of computational physics and engineering [54]. While direct numerical simulation of the Navier-Stokes equations provides high accuracy, its computational demands limit practical applications. Recent physics- informed neural networks (PINNs) [55, 56] have made valuable contributions by embedding physical constraints into learning architectures, though these approaches often involve balancing multiple loss terms and addressing convergence considerations [57].
We demonstrate how Flow-Mixer's constrained architecture offers a complementary data-driven approach on two canonical test cases: two-dimensional flow past cylinder at Reynolds number Re = 150 and flow around a NACA airfoil at Re = 1000 with 15 angle of attack. For 2D fluid dynamics prediction, FlowMixer processes the vorticity field w directly, with the core transformation applied as:
WEIIHA = Φ- (Wioluu
(15)
where wi-++i# represents a sequence of h consecutive vorticity snapshots, and Wh+1:tth is the predicted sequence for the next h timesteps.
tt 28
At 5
- 3,20
5t 6.40
Graund Trush
Flowhiiaur
MBE: 326-03
MBE: 33346-03
MSE: 3.50e-02
MSE 1:52e-03
Eror
Figure 4: Prediction of vorticity fields for flow past cylinder at Re 150 (left columns) and NACA airfoil at Re = 1000 (right columns). (a) Ground truth vorticity fields showing the evolution of wake structures. (b) FlowMixer predictions demonstrate accurate capture of both near-body structures and downstream vortex evolution. (c) Error fields with corresponding Mean Squared Error (MSE) values show minimal discrepancy between prediction and ground truth across varying geometries and flow regimes. Experimental details are provided in Appendix N. Additional flow visualizations and analyses are provided in Appendix N.4 and Appendix N.5.

---

Our simulations capture the full dynamics across computational domains with 400 x 160 resolution. The cylinder flow characterized by the von Kärmán vortex street with a Strouhal number of approximately 0.2, while the airfoil case introduces asymmetric geometry, flow separation, and higher Reynolds number effects. For both cases, we train Fixer to process 64 consecutive flow field snapshots and predict the next 64 timesteps (h- = 64) represennnnn physical time horizons of 6.4 seconds for the cylinder and the airfoil. Figure 4 compares the predicted vorticity fields with ground truth. For the cylinder case, FlowMixer accurately captures the alternating clockwise (blue) and counterclockwise (red) rotation patterns the wake. the more challenging airfoil case, the model successfully predicts the complex separation patterns, asymmetric wake structures, and vortex interactions despite the substantially higher Reynolds number.
Quantitatively, FlowMixer maintains MSE values of approximately 3.3e-3 for the cylinder case and 2e-3 to 8e-3 for the airfoil case, corresponding to approximately 4-5% relative 12 error. These results are competitive with existing approaches in the literature [58, 59, 60), while using standard stochas- tie gradient descent with momentum for training rather than specialized optimization procedures (comparisons of optimization strategies are presented in Appendix N.5).
FlowMixer adapts to increased flow complexity without architectural modifications. When tran- sitioning from the regular vortex shedding of the cylinder at Re = 150 to the higher Reynolds number airfoil case Re- 1000 with complex separation and wake interactions. the model maintains comparable accuracy using the same architectural constraints. The error fields (Figure 4c) show relatively uniform distribution without significant localized spikes, suggesting the model effectively captures dominant flow patterns across different regimes. This behavior is noteworthy given that other approaches often require explicit conservation constraints to achieve stable predictions across varying flow regimes [61, 62, 63].
The successful prediction of complex flow patterns around both simple and complex geometries at different Reynolds numbers demonstrates the versatility of FlowMixer's architecture. The ability to capture these dynamics without specialized physical constraints points toward applications in aerody- namic design optimization, real-time flow monitoring, and reduced- order modeling for engineering systems where both computational efficiency and accuracy are essential requirements.
Conclusion
Flow demonstrates how architectural constraints inspired by dynamical systems theory enhance both predictive capabilities and interpretability neural forecasting systems. Through extensive validation across multiple domains, have shown this constrained approach offers competitive performance in long-horizon forecasting while effectively modeling complex physical phenomena such chaotic attractors and turbulent flows.
The architecture introduces three complementary innovations: (1) constrained non-negative matrix mixing operations, (2) a Kronecker- Koopman framework providing interpretable spatiotemporal patterns, and (3) a Semi-Orthogonn Orthhgonal Basic Reservoir approach enabling stable prediction of chaotic systems. Together, these elements create a unified framework bridging statistical learning and dynamical systems modeling.
Our results highlight an important principle in neural architecture design: carefully chosen mathemati- cal constraints can simultaneously maintain competitive performance while enhancing interpretability without necessitating increased model complexity. Notably, the semi-group composition property eliminates depth optimization entirely: one layer is all you need, This frees practitioners from critical deployment bottleneck that affects all existing architectures.
We acknowledge current limitations, including cubic computational complexity in sequence length and the absence of explicit physical conservation guarantees. Future work techniques to improve computational scaling and investigate integration with physics-informed approaches to enhance scientific applications.
will explore sparsification
As deep learning advances scientific computing, architectures like FlowMixer demonstrate the value of incorporating domain knowledge through mathematical design principles rather than explicit regularization-offee promising direction for neural forecasting systems that are both effective and interpretable across scientific and engineering applications.
10

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Code Availability
The code for FlowMixer available atttte///gghtttooomm//aresssaaap The authors declare no competing interests.
Acknowledgment
This work was supported by the NYUAD Center for Interacting Urban Networks (CITIES), funded by Tamkeen under the NYUAD Research Institute Award CG001. The views expressed in this article are those of the authors and do not reflect the opinions CITIES or their funding agencies
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Question:
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Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing. editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, originality of the research, declaration is not required.
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22

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A Appendix: Long-Horizon Time Series Forecasting
A.1 Experimental Details
We evaluated FlowMixer using seven benchmark datasets encompassing diverse domains with varying temporal resolutions and prediction challenges. The ETT datasets capture electrical transformer measurements over a two-year period, consisting of both hourly (ETTh) and 1 15-minute (ETTm) resolution variants. The Electricity dataset records hourly power consumption from 321 customers, while Traffic contains hourly road occupancy rates from California. The Weather dataset comprises 21 meteorological indicators sampled at 10-minute intervals throughout 2020.
Data preparation followed standard protocols, with chronological partitioning using train/validation/tesst ratios of 0.6/0.2/0.2 for ETT and 0.7/0.2/0.1 for others. Each feature underwent independent z-score normalization. For most datasets we use an input length of 1024 (2720) timesteps. For Electricity and Traffic, we utilize longer input sequences (up to 3072 timesteps) due to their increased feature dimensionality.
Model training was implemented in TensorFlow and executed on single NVIDIA A100 GPU. The training protocol incorporated early stopping with patience of 10 epochs and learning rate reduction (factor 0.1, patience 5) based validation loss. We limited training to 100 epochs with batch size 32 and systematically evaluated dropout rates between 0.0 and 0.7. The model's hyperparameters were optimized for each dataset and prediction horizon combination, with comprehensive configurations detailed in Table 3. While training hyperparameters were optimized per dataset, the core architecture remained fixed with single mixing block- clinknatiinn the depth search required by baseline methods.
A.2 Extra comparison
Table 2: Extra comparison of FlowMixer with N-HITS [64], FEDformer [65], Autoformer [66], Dlinear [67], and some Koopman based models: KooPA [29] and KNF [30]. The reported standard deviations for FlowMixer use five seeds. Bold blue indicate best performance, demonstrating FlowMixer's competitive results.
Flxwr (Ours)
N-HITS (2021)
FEDformer (2022)
Autoformer (2021)
DLinear (2023)
KooPA (2023)
KNF (2023)
h
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96 1011
0.390:1.001
0.475
0.498
0.376
0.415
0.435
0.446
0.386
0.400
0.371
0.405
0.975
0,744 0.744
192
0,394=.000
0.412:t.001
0.426
0.519
0.423
0.446
0.456
0.457
0.437
0.432
0.416
0.439
0.941
336
.01
0.430:t.001
0.550
0.564
0.444
0.462
0.486
0.487
0.481
0.459
720
0.465 1.001
1..75.1.001
0.598
0.641
0,469
0.492
0.515
0.517
0.519
0.516
96
0.264:1.001
0.330-1.001
0.328
0.364
0.332
0.374
0.332
0,368
0,333
0.387
0.297
0,349
0.433
0.446
192
0.320=20002
0.368=1.001
0.372
0.408
0.407
0.446
0.426
0.434
0.477
0.476
0.356
0,393
0.528
0,503
336
0.344:2002
0,396:t.001
0.397
0.421
0,400
0.447
0.477
0.479
0.594
0.541
720
0.420
0.452.1.002
0.461
0.497
0,412
0.469
0.453
0.453
0.831
0.657
96
0.298.1.00
0.345:1.002
0,370
0.468
0.326
0.390
0,510
0.492
0.345
0.172
0.294
0.345
0.957
0.782
ET1m2 ET1ml
192
0.333-1001
0.368-1.011
0.436
0.488
0.365
0.415
0.515
0.514
0.380
0.389
0.337
0.378
0.896
0.731
336
0.363=#001
0,383:t.001
0.483
0.510
0,392
0.425
0.510
0.492
0.413
0.413
720
0.404:1.001
0.410:1.001
0.489
0.537
0.446
0.458
0.527
0.493
0.474
0.433
96
0.01
0.252.1.001
0.176
0.255
0,180
0.271
0.205
0.293
0.192
0.282
0.171
0.254
1.535
1.012
0.211:1001
0.290-1.001
0.245
0.305
0.252
0.318
0.278
0.336
0.284
0.362
0.226
0.298
1.355
0.908
336
0.260=
0,324=t001
0.295
0.346
0,324
0,364
0.343
0,379
0,369
0.427
720
0,333:10001
0.377:1.001
0.401
0.413
0.410
0,420
0.414
0.419
0.554
0.522
Weather
96
0.143.1.000
0.19411.002
0.158
0.195
0,238
0.314
0.249
0,329
0.193
0.292
0.154
0.205
0.295
0.308
0.005
0.238-L000
0.211
0.247
0,275
0.325
0.325
0.325
0.228
0.296
0.193
0.241
0.462
0.437
336
0.235=2002
0,276:t.001
0.274
0.300
0,339
0.377
0.397
0.397
0,283
0.335
720
.001
0.326:1.002
0.351
0.353
0.351
0.409
1.042
1.216
0.345
0,381
Electricit
96
0.131 10001
.003
0.147
0.249
0.186
0.302
0.196
0,302
0.197
0.282
0.136
0.236
0.198
0.284
0.006
0.23911.003
0.167
0.249
0.197
0.311
0.211
0.321
0.196
0,285
0.254
0.245
0.321
336
0.160= 000
0.255:1.002
086
0.290
0,213
0.328
0.214
0.328
0.209
0.301
720
0.195-:.00 .003
0.289-10004
0.243
0.340
0.236
0.344
0.236
0.342
0.245
0.333
96
0.377 .002
0.264:1.001
0.402
0.282
0.576
0.359
0.597
0.371
0.650
0.396
0.401
0.275
0.645
0.376
0,388 1.002
0.268 1.022
0.420
0.297
0.610
0.380
0.607
0.382
0.598
0.370
0.284
0.699
0.405
336
0.401:1002 .002
0.275
001
0.448
0.313
0.608
0.375
0.623
0.387
0.605
0.373
720
.43
0.291
0.539
0.353
0.621
0.375
0.639
0.395
0.645
0.394
23

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B Periodicities Analysis
Pariod (hound)
Period (houn))
ETTma
ETTm2
124
Mognttdte
Sincc
33
Period (houn))
Period (houn)
Weather
Electricity
127
Megritdee
Sracely
Period
Traffic
Exchenge
12
53.h
MMM M
Dacaky
044
32
Period (rouin))
Pariod (oous))
Figure 5: FFT Spectral Analysis of Periodicity Patterns in Benchmark Time Series Datasets. The plots show magnitude versus period (in hours) on logarithmic scales for eight standard forecasting datasets. Significant periodicities are marked with red dots and vertical lines, revealing prominent daily (24h) patterns across all datasets, with additional weekly (168h) and biweekly cycles of varying strengths. These spectral characteristics inform our model design, particularly for incorporating appropriate time mixing components and periodicity-awan architectures capture multi-scale temporal dependencies. We select mainly periods (24,168) our experiments.
24

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C Appendix: Hyperparameters for Experiments
Table 3: Hyperparameter configurations for different forecasting horizons (h) and input lengths (Input Len.). Parameters include optimizer choice (Optim.), initial learning rate (Init. LR), weight decay (W. Decay), RevIN type, time per epoch (T/Epoch), exponentia/qquadraii time mixing (Expm), and periodicities. "Turb." stands for turbulence.
Input Len. Batch Optim. Init. LR W. Decay
RevIN Time/Epoch Expm
Periodicities
96
1344
16
SGD
le-1
le-6
RevIN
2s
True
[24. 168]
192
1344
SGD
le-1
lc-6
RevIN
True
124, 168]
336
1344
SGD
le-1
le-6
RevIN
2s
True
124, 168]
720
1344
16
SGD
le-1
Ic-6
RevIN
2s
True
[24, 168]
96
1344
16
SGD
le-1
lc-5
RevIN
2s
False
Traffic Electricity Weather ETTm2 ETTml E7Th2
192
1344
16
SGD
le-1
le-5
RevIN
2s
False
336
1344
16
SGD
le-1
le-5
RevIN
2s
True
[24, 168]
720
1344
SGD
le-1
le-5
RevIN
True
[24, 168]
96
1344
AdamW
lc-3
Ic-6
TD-RevIN
2s
False
1344
AdamW
le-3
le-6
TD-RevIN
True
[24 4, x 4] [24 4, 168 x 4]
336
1344
16
A
le-3
le-6
TD-RevIN
2s
True
720
1344
16
AdamW
le-3
le-6
TD-RevIN
2s
True
[24 4, 168 x
96
1344
16
SGD
le-1
Ic-6
RevIN
2s
Falsc
192
1344
16
SGD
le-1
lc-6
RevIN
28
Falsc
336
1344
16
SGD
le-1
le-6
RevIN
2s
False
720
1344
SGD
le-1
le-6
RevIN
False
1344
AdamW
le-3
le-6
TD-RevIN
False
192
1344
A
le-3
le-6
TD-RevIN
8s
False
336
1344
16
AdamW
le-3
1c-6
TD-RevIN
8s
False
720
1344
16
AdamW
le-3
1c-6
TD-RevIN
8s
True
96
2688
16
AdamW
le-4
le-6
RevIN
122s
False
192
2688
16
AdamW
le-4
le-6
RevIN
122s
False
336
2688
AdamW
le-4
le-6
RevIN
122s
False
720
2688
AdamW
lc-4
Ic-6
RevIN
122s
Falsc
4032
Adam
le-3
le-6
TD-RevIN
98s
False
192
4032
AdamW
le-3
le-6
TD-RevIN
98s
False
[24, 168]
336
4032
16
AdamW
le-3
le-6
TD-RevIN
98s
False
[24. 168]
720
4032
16
AdamW
lc-3
lc-6
TD-RevIN
98s
Falsc
Turbu. Chaos
32
AdamW
le-3
1e-6
TD-RevIN + SOBR
False
64
64
32
SGD
le-1
le-6
RevIN
28s
False
The hyperparameters for our time series forecasting experiments were determined through system- atic grid search across multiple dimensions. We evaluated two normalization approaches (RevIN and TD-RevIN) to address distribution shifts in time series data. For optimization strategies, we explored both AdamW W (with learning rates of 0.001 and 0.0001) and SGD with momentum 0.9 (using learning rates of 0.001, 0.01, and 0.1). Additional hyperparameters included various batch sizes, weight decay values (1e-5, le-6), and different periodicity configurations based on the inherent temporal patterns each dataset. Notably, the use of periodicities and matrix exponential transformations was primarily implemented for the ETT datasets, as the computational cost was prohibitively high for larger datasets such as Electricity and Traffic. This comprehensive search allowed us to identify the optimal configuration for each forecasting horizon and dataset, balancing computational efficiency with forecast accuracy. Note: Optimizer choice (SGD vs Adam W) was determined empirically for each dataset, but the core architecture remained consistent across all experiments.
25

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D Statistical Validation: Wilcoxon Signed-Rank Tests
We conducted Wilcoxon signed-rank tests across all 28 dataset-horizon combinations validate the statistical significance of FlowMixer's improvements over baseline methods. The Wilcoxon test was chosen makes no assumptions about the distribution of improvements and is robust to outliers, making it ideal for comparing forecasting models across diverse datasets with potentially non-normal error distributions.
Table 4: Statistical evaluation of FlowMixer against baseline methods
Metric
Chimera
TimeMixert+
iTransformer
Ada-MSHyypr
CycleNet
TSMixer
Mean Squared Error (MSE) Analysis
W-statistic
119.5
0.0 <0.001
0.0
1.0
0.0
38.0
P-value
0.047
<0.001
<0.001
<0.001
<0.001
Significant (a=0.05)
V
v
v
Wins 28
28
28
Win Rate
64.3%
100%
100%
96.4%
100%
82.1%
Avg. Improvement
0.93%
4.98%
12.78%
6.20%
10.62%
3.17%
Mean Absolute Error (MAE) Analysis
W-statistic
135.5
171.5
2.0
112.0
2.0
12.0
P-value
0.099
0.236
<0.001
0.032
<0.001
<0.001
Significant (a=0.05)
X
V
V
V
Wins / 28
18
14
27
21
26
20
Win Rate
64.3%
50.0%
96.4%
75.0%
92.9%
71.4%
Avg. Improvement
0.89%
0.24%
6.02%
1.20%
4.61 %
2.12%
Combined MSE+MAE Analysis (56 comparisons)
W-statistic
497.0
288.0
2.0
194.5
92.0
P-value
0.017
<0.001
<0.001
<0.001
<0.001
<0.001
Significant (a=0.05)
V
Wins 56
42
55 98.2%
Win Rate
64.3%
75.0%
85.7%
96.4%
76.8%
Avg. Improvement
0.91%
2.61%
9.40%
3.70%
7.61%
2.64%
Overall Performance Summary
Methods Outperformed
MSE: 6/6
MAE: 4/6
Combined: 6/6
Average Win Rate
90.5%
75.0%
82.7%
Average Improvement
6.45%
2.51 %
4,48%
Note: Wilcoxon signed-rank test with one-sided alternative (H1: FlowMixer < Baseline). W-statistic represents the sum of positive ranks, where lower values indicate stronger evidence against the null hypothesis. Significance threshold: a = 0.05.
FlowMixer achieves statistically significant improvements (p<0.05) over all baseline methods for MSE, confirm- ing that the performance differences not duc to random variation. While the magnitude of improvements is incremental (1-12%), these gains meaningful in the context of mature benehmarks. The W-statistic values near for several comparisons (TimeMixert+, iTransformer, CycleNet) indicate that Fixer outperformed these methods on nearly all tataset-horizon pairs. While MAE shows non-significance against Chimera and TimeMixer++. this is expected as MAE is less sensitive to extreme errors that MSE captures and and extreme prediction crucial for practical forecasting applications. The average win rate of 82.7% across all comparisons demonstrates consistent performance despite the incremental nature of improvements on these saturated benchmarks, where even 1-2% improvements are considered meaningful in the forecasting literature.
26

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E Further Details on the FlowMixer Architecture
E.1 Mixing Expressions
W, Soft (ax(QK" /vde)
Features Mixing
1 12
Mixing Layer
X
ReviN/DDS norm
Φ (X)
W. (
RevIN/50B2 denorm
y (2)
Time Mixing
aI W x W
Standurd,
W. Ep Wr(p) @ Wp, p r (p)
(Wox)W)
Exponential,
Periodic +
e.g.14 = 1344, = 24, (p) = 1344/ 24 56
al Wrp x Wro Periodic,
e.g.4 = 1314, p= = 21 x 7, (p) 1344//168
Exponential Periodic,
Figure 6: Overview of the FlowMixer architecture. Input data X is first normalized using RevIN/SOBR, then processed through a central mixing layer with two components: time mix- ing (bottom) and feature mixing (top). The mixing layer transforms the normalized input ((X) using weight matrices We and W,, resulting in an intermediate representation Z that is finally denormalized to produce the output Y. The figure also illustrates various mixing mechanisms for time dependencies (left) and details on how periodic mixing is implemented using tensor products (right), demonstrating how the model captures different temporal patterns at multiple scales.
E.2 Selection of Input Sequence Length
The selection of appropriate sequence length nt, is critical for capturing meaningful temporal patterns in time series forecasting. For FlowMixer, we establish that the lookback window should longer than the maximum prediction horizon (720), thus n, > 720. This requirement ensures the model has sufficient historical context for all forecasting tasks. Based on our spectral analysis, we identified that the most prominent periodicitics in the ETT datasets are daily (24 hours) and weekly (24 x 7 168 hours) cycles. For ETTm datasets specifically, these periods multiplied by factor of four due the 15-minute sampling rate compared to hourly sampling other datasets. To accommodate both these periodic patterns and allow for effective multiplicity search in our structured decomposition, we selected nt, = 1344 = 24 x x 4 x 2. This value represents two complete cycles of the largest periodicity (biwecklyy in the 15-minute sampled data. For Electricity data, we increased the sequence length by factor of (n, 2688) to improve results, particularly for longer horizons where capturing multiple eycles of longer-term patterns proved beneficial. Similarly, for Weather data, increased ni by factor 3 better model complex atmospheric dynamics exhibit longer-range dependencies.
E.3
Chaneee-Indeeendentt Flow Mixer Variant
E.3.1
Mathematical Formulation
The channel-independent variant decomposes multivariate forecasting into ny independent univariate problems. Definition E.1 (Channel-Indeppendenn FlowMixer). For input X E "R","", the channcl-independent operation
is:
Y.₁ = (X..)) Vj e [1,
(16)
where cach F, :. + Rd operates independently:
Fj(z) = (We,
(17)
E.3.2 Theoretical Properties
Proposition E.2 (Properties of Channel-Independee Variant). The channel-independer FlowMixer preserves:
27

---

1. Semi-group property (per chunnel): F3 F3 -
2. Temporal eigenmodes: Each channel maintains independent temporal decomposition
3. Algebracc horizon modification: Per-channel via (W h
But loses:
Kronecker structure: pyatial-emmpord coupling
2. Cross-channel dependencies: Connot model feature interactions
E.3.3
Computational Analysis
Table Parameter count comparison
Configuration
Standard Flow
Ghannel-Independers
Time mixing
T2 n2 n n
ny x n 0 n X n
Feature mixing
Total parameters
ETTh1 (7 channels)
1.8M
12.7M (7x) 1, 1,566M (489x)
Traffic (862 channels)
3.2M
E.3.4 Empirical Performance
Table 6: Performance comparison on ETThl (MSE)
Horizon
192
336
720
Standard FlowMixer
0.358
0.394
0.418
0.465
Channel- Leneeee-Ineeenndent
0.371
0.412
0.439
0.491
PatchTSTT (channel-ind.)
0.370
0.413
0.422
0.447
Remark E.3 (Trade-offis.
The channel Ghaanel-independen variant:
Gains: Per-channel optimization flexibility, parallel training
Loses: Cross-channel patterns, parameter efficiency
Optimal when: Channels are truly independent (e.g.. unrelated sensor streams)
Suboptimal for: Spatially correlated data (turbulence, traffic networks)
FlowMixer's target applications (traffic, turbulence) where spatial correlations are fundamental, standard architecture with full mixing matrices proves more suitable despite higher computational complexity.
28

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Appendix: Flow.Mixer Ablation Study
To understand the contribution of each component in FlowMixer, conducted a comprchensive ablation study the ETThl dataset. Figure 7 illustrates the comparative performance analysis of different model variants, while Table presents the detailed numerical results.
We tested the following architectural variants:
FlowMixer (Full): The complete model as described in Methods and the supplementary material.
w/o RevIN: FlowMixer without Reversible Instance Normalization.
w/o Feature Mixing: FlowMixer with the feature mixing component removed.
w/o Time Mixing: FlowMixer with the time mixing component removed.
w/o Positive Time Mixing: FlowMixer without the positivity constraint time mixing.
w/o Static Attention: FlowMixer with a generic mixing structure for features instead of the proposed
approach.
w/o Adaptive Skip Connection: FlowMixer without the adaptive skip connections.
FlowMixer Abladon Study on ETTh1 Dataset
(a) Ab lotooio Study MIE vs. Forecast Horizon
(b) Average Performance Across Honizons
0.65
Plunffacc thitt
0.575
0.96
0.60
Mhong Al Pealive Time Miig All arantion
0.525
0.56
MSE
0.500
0.50
6.475
0.481
0.45
0.400
2:428
0.40
0.455
0.822
0.50
0.400
0.35
100
Paroo Mposkbong
Adaaso bkly Cansadionn
Redure Merogg
Tamt Aeeutio
lerell
Medee
Model Vanant
Figure 7: Comprehensive ablation study of Fixer architectural components. a. MSE perfor- mance across multiple prediction horizons (48-720 hours) for different model variants. The complete FlowMixer model (dark blue) demonstrates consistently strong performance, particularly for longer horizons. b, Relative performance degradation (%) quantifying the impact of removing individual components. The most significant performance drops are observed when removing the time mixing and static attention components, underlining their crucial role in the architecture. Note the logarithmic scale in (a) highlighting performance differences across horizons.
Table Ablation study results on ETThl dataset. We report average MSE for different prediction horizons, with + std from five seeds.
Model Variant/ Horizon
48
168
336
720
FlowMixer (Full)
0.336 0.001
0.358 0.001
0.394 0.011
0.418:: 0.001
0.465 0.001
w/o RevIN
0.341 00003
0,371 0.003
0.412:: 0.022
0.461 0.066
0.006
0.620 0.012
w/o Feature Mixing
0.378 0.04
0.390 0.001
0,410 0.001
0.427 0.022
0,476 0.002
w/o Time Mixing
0.486+00014
0.523 0.088
0.543 0.005
0.555
0.634= 0.009
w/o Positive Time Mixing
0.338
0.3644 0.001
0.398 0.002
0.427 0.001
0.482
w/o Static Attention
0.373
0.386 0.007
0.413 0.00s
0.4322+ 0.o6s
0.519 0:007
w/o Adaptive Skip Connection
0.336 0.002
0,362 0.001
0.395 4 0.001
0.420 0.001
0.497 0.001
The results demonstrate several key insights about FlowMixer's architecture:
29

---

Time Mixing;
The time mixing component proves essential, with its removal leading to the most substantial performance degradation (up 36% relative increase in MSE). mainly affecting longer- horizon predictions.
The static attention mechanism for feature mixing significantly contributes to model performance, with its replacement by generic mixing structure resulting in inconsistent performance across horizons.
Feature Mixing:
RevIN: Reversible Instance Normalization plays a crucial role in handling distribution shifts, showing increased importance longer horizons (720h), where its removal lcads to 33.3% performance degradation.
Positive Time Mixing: The positivity constraint on time mixing has minimal impact on performance (around 1.9% degradation on average) but is crucial to reach SOTA.
Architectural Choices: Both adaptive skip connections and dropout contrihute to model rohustness, though their impact is more subtle than the core mixing components. Nevertheless, their contribution is essential reach SOTA.
This comprchensive ablation study validates the architectural choices FlowMixer, demonstrating that each component contributes mcaningfully to the model's overall performance. The results highlight the importance of the time mixing and feature mixing mechanisms, which form the core of FlowMixer's architecture, presented the main text.
G Example of Forecasting Curves
HUFL
b
HULL
10
-20
-36
24
42
12
36
48
84
96
-36
-24
-12
12
24
48
60
72
96
MUFL
MULL
value
-20
-36
-24
-22
12
24
36
48
60
72
84
36
-36
-24
-32
12
24
36
48
50
72
84
96
Time (hours)
Timaa (hours)
Ground Truth
FlowMiixer
NHITS
TSMixer
Figure 8: Comparison of forecasting performance for FlowMixer. NHiTS, and TSMixer on the ETThl dataset. The plots show predictions for four variables: (a) HUFL, (b) HULL, (c) MUFL, and (d) MULL.. The x-axis represents time in hours, with O marking the start the prediction period. The y-axis shows the values for each variable. Ground truth (dashed black line) is plotted alongside predictions from Flow (blue), NHiTS (green), and TSMixer (orange). FlowMixer demonstrates competitive performance, consistently producing forecasts that are more closely aligned with the ground truth across four variables. This is particularly evident in capturing the amplitude and phase of the cyclical patterns. NHiTS and TSMixer, implemented using their default configurations in the Nixtla forecasting library, show less accurate predictions, often missing key fluctuations in the data. FlowMixer's 's enhanced predictive capability is especially notable in periods of rapid change unusual patterns, suggesting a more robust and adaptive forccasting approach.
30

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Algebraic Horizon Modification: Theory and Practice
H.1
Overview
FlowMixer's s algebraic horizon modification capability represents a fundamental departure from traditional forecasting approaches. Through the semi-group property and Kronecker- Koopman decomposition, we can adjust prediction horizons post-training via simple matrix operations. This section provides theoretical foundations, practical demonstrations, and comparisons with existing Koopman- methods including Koopa [29] and
KNF [30].
H.2
Practical Illustration
Performing horizon change is straightforward using equation 12. Our theoretical framework allows both interpolation (f < 1) and extrapolation (t > 1) of horizons through eigenvalue scaling:
X(h') x-
Σ
(ap3 (Wnng F2)
(18)
To demonstrate this capability and its limits, conduct experiment on ETTh1 with model trained for horizon h then algebraically adjust predict various other horizons.
FlowMixer Horizon Modification ETTh1
Horizon: hours
Horizon: hours
Coxed Trth
VAMe
Smant ktup (HUSS)
Horizon: hours
Herizon:
Dlvember 2011
Goonnd Hauetion mattop (10aa
Horizon:
Grrand trath
-30
Pluntiborr Outrng (14537
-25
Figure 9: FlowMixer trained for horizon 96 (green), then algebraically adjusted for various horizons. The model maintains reasonable performance for e 2.0), with degradation beyond this range. Values closer to 1 yield best results naturally, as the model was optimized for the original horizon.
results reveal that FlowMixer can successfully adjust horizons within a factor (halving or doubling) while maintaining reasonable accuracy. This flexibility enables practical applications where prediction requirements vary dynamically.
H.3 Theoretical Error Bounds
We now establish theoretical foundations for understanding extrapolation performance and its limitations.
Theorem H.1 (Horizon Extrapolation Error Bound), For FlowMixer F with time mixing spectral radius p(W.). feature mixing spectral radius p(W,), and RevIN condition nunber max, [nyl/aj the k-step prediction minn 1731795
31

---

Interpollatoo//xxrrpolaaao
20.0
Mear: Squated
Interpolation
Extrapoiation
8.0
interpabation (hel) Eetropolation (tx2)
0.6
Prodiction Horizon
Figure MSE for interpolatedeextraplaaeed horizons on log scale, showing approximately exponen- tial error growth for large deviations from the training horizon.
ervor satisfies:
-x) S p(W,) P(W))
(19)
where "> are RevIN's learned affine parameters and o are instance-wise standard deviations.
Proof. Lo X. X + and denote X; x. F((i-))
Step 1: Single-step error bound. For one FlowMixer application:
|& X1ll = 1 (WWo)((()) (
(20)
< [|aa ')ll [| - Wio(X.)W7
(21)
W1
Fideoom W (WW)o( (ox))))) - ((Xo)l
(22)
Finemmm (W. (Wi)
(23)
where Naoem = max, h7sl/ag bounds the normalization Jacobian and Kidenum max, ////s bounds the
denormalization Jacobian.
Step 2: Multi-step proaggation. By induction on k: For h: = 2:
X2 X2ll S Kennorm (W,, (WW)) -(xx))))
(24)
S Kemnrr (W (WW Kinom 11 X1ll
(25)
S x² p(W.) p(W,) [|cl)
(26)
The general case follows by applying this argument recursively:
Ix Xull K² p(W.)* M[ll
(27)
Sinec Ngorm Kidenomm the bound follows.
Corollary (Stability Conditions for Extrapolation). For stable long- long-horizon extrapolation, we require:
p(W,) S and p(W,) (spectral normalization)
2. k. R (well-conditioned normalization)
3. Small initial perturbation [|e]]
H.4
Implications and Design Considerations
The error bound reveals three critical factors affecting extrapolation:
Spectral radii control: Even with p(W.) p(W,) = 1, errrr can grow as *
RevIN conditioning: The normalization's 's condition number fundamentally limits extrapolation range
Eigenvalue constraints: For improved stability. could enforce:
W, = exp(A )") (orthogonal, all cigenvalues on unit circle)
(28)
This preserves energy but may reduce expressivity.
32

---

H.5
Comparison with Koopman-Basee Methods
We compare FlowMixer's algebraie approach with existing Koopman- boopman-based forccasting methods, particularly Koopa [29] and KNF (Koopman Neural Forecaster) [30].
H.S.1 Methodological Differences
Koopa [29] addresses non non-stationary time series through learned Koopman operators but requires operator adaptation (OA) for horizon changes. Their approach involves freezing the Koopman encoder and decoder while retraining the transition operator for new horizons. In contrast, KNF [30] handles temporal distribution shifts through implicit Koopman framework hut lacks explicit horizon adjustment mechanisms.
FlowMixer's approach differs fundamentally: manipulation of horizons without any retraining or architectural modifications.
Kennecker-Koopmene structure enables direct algehraic
Table 8: Forecast performance comparison: baseline 48 and scale-up to h 144 using different horizon adjustment methods. FlowMixer's algebraic adjustment outperforms Koopa-based methods this evaluation.
ETTh2
ECL.
Traffic
Weather
Setting
Method
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
Baseline
FlowMixer
0.264
0.330
0.131
0.226
0.377
0.264
0.143
0.194
(h 48)
Koopa [29]
0.297
0.349
0.148
0.240
0.395
0.268
0.174
0.214
Flowrixer (algebraic)
0.339
0.385
0.151
0.246
0.397
0.277
0.191
0.242
Scale-Up (h 141)
Flow Floer (retrained)
0.323
0.362
0.146
0.239
0.392
0.271
0.188
0,238
Koopa (rollout) [29]
0.437
0.429
0.199
0.298
0.709
0.437
0.237
0.276
Koopa+OA [29]
0.372
0.404
0.182
0.271
0.699
0.426
0.225
0.264
H.6
Theoretical Comparison
Table Theoretical properties Koopman- based forecasting methods
Property
FlowMixer
Koopa [29]
KNF [30]
Semi-group property
Guaranteed Direct Algebraic No ((1)
Approximate
Not enforced
Eigenmode extraction
Embedded
Implicit
Horizon flexibility
Operator adaptation
Fixed
Retraining needed
Partial
Full
Computational cost
O(epochs)
Ofull training)
H.7
Method Implementation Comparison
Table 10: Implementation details for horizon adjustment
Method
Mechanism
Retraining
Time
Flexibility
FlowMixer
Matrix power Whh /h Iterative application
None
1s Continuous
Koopa [29] (rollout)
None
O(h)
Integer only
Koopa+OA [29]
Operator adaptation
Partial
Minutes
Integer only
KNF [30] Retrain from seratch
Full retraining
Complete
Hours
Any horizon
Full training
Complete
Hours
Any horizon
H.8 Key Observations
Performance: FlowMixer's algebraic adjustment outperforms both Koopa [29] with or without
operator adaptation and KNF [30] when horizon adjustment is required
Efficiency: Matrix powering takes milliseconds versus minutes for Koopa's operator adaptation or hours for KNF's full retraining
33

---

Flexibility: FlowMixer enables fractional horizons (e.g.. h 324.5) impossible with integer-based
rollout methods in Koopa
Simplicity: No architectural modifications or retraining required, unlike both Koopa and KNF
Theoretical grounding: Error bounds provide principled understanding of extrapolation limits.
I Complexity Analysis and Stability
L.1 Computational Complexity
Understanding the computational demands FlowMixer helps assess its practical applicability across different domains. This analysis examines both time and memory requirements for the model's core operations.
Time Complexity. For an input matrix E R where n, represents the number of time steps and ny the number of features, Mixer's computational demands stem primarily from three operations:
The time mixing process begins with forming the matrix W2 = al W x W, requiring O(n²) operations, but importantly, this computed once per forward pass rather than per feature. The subsequent matrix multiplication W, X involves multiplying the pre-computed W. with each feature column, incurring cost of ((nīn,). Using
the exact matrix exponential W, (Wo would significantly increase computational cost to O(m)). which motivates our first-order approximation.
When employing the Kronecker structure for seasonality, W, decrease providing some efficiency gains for long sequences with periodicity.
E, Wr()) Wp. the computational cost
Feature mixing through static attention computes key-query interactions QK" /va d the attention dimension. The subsequent SoftMax and matrix multiplication add (nn me) operations. These computations enable the model capture feature interdependencies.
O(n} d) cost, where d is
The RevIN component performs normalization and denormalization with O(nery) operations, addressing distribution shifts between training and inference.
Combining these components yields an overall time complexity bound
T(me, ny, d) On2n, , nin, n d)
(29)
When time steps and feature dimensions are comparable (nt ze ny = n) and d < n, this simplifies to approximately On="). ). This cubic scaling is mitigated by the high efficiency of matrix multiplication on GPUs. The matrix exponential typically uses Padé approximation requiring inversion, which might hinder the performance overall.
Memory Complexity. FlowMixer's memory requirements from storing the mixing matrices and intermediate activations, The time mixing matrix W. E R and feature mixing matrix W, x require (n and On??) storage, respectively, Intermediate activations add an additional (n.ny) memory requirement. Combined, the memory complexity is:
M (m., ny) = On1 + + + +
(30)
With Kronecker structure for seasonality modeling, memory requirements for W. decrease to O(2 + +²) where r x p = ne, substantially reducing storage needs.
Scalability Strategies. To address computational challenges with large-scale data, Fixer employs several optimization strategics:
First, low-rank approximations can factorize W, or W, into rank-r products, reducing computational require- ments to Orr(n, + n2))). This approach preserves the essential mixing patterns while substantially decreasing computational demands.
Second, sparse attention mechanisms can restrict feature mixing block-sparse local patterns. This reduces the effective complexity to O(n, 1y) making the model more efficient for high-dimensional feature spaces.
Third, chunked processing divides long sequences into blocks size k. lowering the per-block complexity to Okknny). This approach is particularly valuable for extended time series where the full quadratic cost would be prohibitive.
These optimization techniques enable FlowMixer to scale efficiently to practical applications with long sequences nigh-dimensiooal feature spaces while maintaining its theoretical properties.
34

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1.2
Stability Analysis
FlowMixer's numerical stability derives from carefully designed spectral properties in its mixing matrices, ensuring robust performance across diverse forecasting tasks.
I.2.1
Feature Mixing Stability
The feature mixing matrix Wy is constructed as left-stochastic matrix through SoftMax- based operation. This design ensures that its maximum eigenvalue is exactly one (Amax 1), effectively preventing unbounded amplification during feature propagation. This property creates natural stability constraint that helps the model maintain numerical robustness even for complex, multi-dimensional datasets.
1.2.2
Time Mixing Stability
The time mixing matrix W, = al + (W W) incorporates positivity constraint through the Hadamard square operation. According the Frrron-Frobeiuus theorem, this structure guarantees dominant positive eigenvalue, contributing stable temporal propagation. For applications requiring fully conservative dynamics, one could normalize W. by its maximum eigenvalue X impose unitary constraints.
Notably, FlowMixer deliberately omits strict normalization allow the model capture both conservative and dissipative dynamics- -a crucial capability for modeling real-world systems where energy may be gained or lost over time. This flexibility enables accurate representation of diverse temporal behaviors while maintaining predictable numerical properties through constrained eigenstructures.
1.3
Computational Scaling Analysis
L.3.1 Empirical Wall-Clock Runtime Analysis
We empirically evaluated Fixw'' computational scaling understand practical deployment boundaries beyond thcoretical complexity.
Table 11: FlowMixer wall-clock runtime scaling (ETTml, pred_len=96, A100 GPU)
Sequence Length
256
512
1024
2048
4096
8192
16384
Time/1000 iter (s)
2.01
2.13
2.33
4.05
8.53
22.5
77.1
Theoretical (n
1.00
4.00
16.0
64.0
256
1024
4096
Actual Scaling Factor
1.00
1.06
1.16
2.01
4.25
11.2
38.4
GPU Elficiency
100%
94%
86%
49%
23%
8.7%
2.3%
Remark 1.1 (Sub-quade quad-uuaccticc Behavior). FlowMixer exhibits sub-quadratic scaling to 4096 timesteps due to efficient GPU parallclization of matrix operations. Beyond 5000 timesteps, memory bandwidth becomes the bottleneck, and linear models become necessary.
1.3.2 Comparison with Linear-Complexity Models
Table 12: Complexity comparison across architectures
Model
Complexity
Key Properties
FlowMixer
Onfin + + O(n))
Full eigendecomposition, interpretability Optional, for dynamical properties Linear scaling, diagonal approximation
FlowMixer (expm)
S4/Mamba
O d k)
Transformer
O .d)
Similar to Flow Mixer
Linear Attention
O(n d²)
Kernel approximation
FlowMixer is computationally advantageous when:
1. Sequence length nı < 5000 (sub-quadratic GPU regime)
2. Interpretability via eigendecomposition is required
3. Single-mode model deployment
multiple datasets is needed
Algebraic horizon manipulation beneficial
For 10000, limear-co models become necessary.
35

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1.4 Matrix Exponential Approximation Analysis
The time mixing matrix W; exo Wo) connects to Koopman theory where temporal evolution is naturally exponential. We investigate the trade-off between computational efficiency and accuracy when approximating this exponential using Taylor series truncation:
exp(A)
(31)
k!
2
k=
Table 13: Performance comparison of Taylor approximation orders (ETThl dataset)
Order
Order 2
Order
Order
Order
Exact
MSE (h=96)
0.362
0.361
0.362
0.361
0.361
0.356
MSE (h=720)
0.488
0.479
0.475
0.470
0.469
0.471
Runtime (s/epoch)
1.8
2.0
2.0
2.0
2.1
5.8
Relative Cost
1.0x
1.11x
1.11x
1.11x
1.17x
3.22x
Key Finding: Higher-order approximations provide negligible accuracy gains (<. 1% MSE improvement) while the exact exponential increases runtime hy 3.22x. GPU parallelization dominates computation time, making orders 1-5 nearly equivalent 11.8...ls). We adopt the first-order approximation W; = a/ + W. x Wo for its simplicity and efficiency--ue the quadratic term already provides sufficient nonlinearity for temporal dynamics.
Theoretical Connections to FARIMA Models
This section establishes connection between FlowMixer's adaptive skip connections and fractional autore- gressive integrated moving average (FARIMA) models.
In ARIMA(p.d.q) models, first-order integration (d=1) transforms non-stationary series into stationary ones through differencing:
VX. X, X,-1
(32)
Neural networks implement conceptually similar operations through residual connections:
F(x,, 0)
(33)
FlowMixer makes this integration process learnable through parameter
#++1 = car F(x,, 0)
(34)
Rearranged with the lag operator L:
(1 aL)xe+1 F(x,,
(35)
This formulation parallels FARIMA models, where fractional differencing operator has first-order Taylor expansion:
(1 = O
(36)
Our a parameter functions analogously to d, controlling memory persistence. The key innovation is making & learnable, allowing the model to adaptively determine appropriate memory persistence for each dataset. This approach offers two primary advantages over fixed fractional differencing:
Automatic Parameter Tuning: FlowMixer learns the optimal a directly from data rather than requiring a priori estimation
2. Computational Tractability:
The first-order approximation maintains efficiency while capturing essential long-memory characteristics
Through this mechanism, FlowMixer effectively addresses non-statinnarity through theoretically grounded, differentiable approach inspired by classical time series models.
36

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K
Chaotic Systems Prediction
K.1
Overview
We evaluated FlowMixer three canonical chaotic systems with distinct dynamical characteristics. Each system defined by of ordinary differential equations:
Lorenz System.
[o 10.0, 3 8/3, p 28.0. Lyapunov exponent 0.91]
oly x)
x p 2) -
(37)
zy Bz
Rössler System.
[a 0022, 0.2, 5.7]
i = y 2
i + ay
(38)
+ 2(x C)
Aizawa System.
[a 0.95, 6 0.7, 0.6, d 3.5. 0.25, f 0.1]
(z b)ue dy
ý dr (a b)y
(39)
c + az
- (a y y² cz) + 1
Data Generation Protocol.
Trajectories were generated via order Runge- Kutta integration (dt 0.01s)
with the following specifications:
Initial conditions: [1.0, 1.0, 1.0] with 500-step transient removal
Sequence length: 12,500 timesteps (125 time units)
Normalization: Min-max scaling to []
Dataset partitioning: 70% training, 15% validation, 15% testing (chronological)
Model Configuration. Flow with SOBR configured with:
Time/feature hyperdimensions: 1024/64
Leaky ReLU activation (a 0.1)
Optimization: AdamW (Ir 10-4, weight decay=10
Training: 100 epochs, early stopping (patience=10)
Comparative Methods. Benchmark models included:
Reservoir Computing: 300 nodes, spectral radius 1.1, connectivity 0.05
N-BEATS: 30 blocks, 3 stacks, 256 units per layer
Performance evaluation covered 1024 prediction steps (~-9-10 Lyapunov times for Lorenz) using both trajectory alignment metrics and attractor reconstruction analysis. This framework assessed each model's capacity to capture complex dynamics characterize these systems across different attractor geometries.
37

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K.2
Semi-Orthogonal Basic Reservoir: Mathematical Foundations
K.2.1
Semi-group Preservation Through Koopman Lifting
The Semi-Orthon Semi-Orthogonal Basic Reservoir (SOBR) addresses the challenge of modeling chaotic systems where low-dimensional uonlinear dynamics require high-dimensional representations for linear approximation.
Proposition K.1 (Semi-group Preservation in Lifted Space), Consider the standard FlowMixer operation:
F(X) o (WW(((()))
(40)
With SOBR, we introduce a lifting operator S R + Rdexds where dy > ny and dj > nz. Let
f St F S be the FlowMixer network with SOBR included.
We work in spaces:
1. Original space (X,Y) Y) with F as an approximate mapping between X and mapping. For example 1e loss could expressed: Esmadand [|
2. Lifted space (S(X),S(Y) with J as approximate mapping between S(X) and S(Y). This choice impacts the loss functions, for an 12 loss we get: Engiet = [[(((())) 3(
the lifted space, the semi semi-group group property is preserved:
Folh) F(h) F((+th) in Rdxdd
(41)
Proof. In the lifted space Rdxxd, FlowMixer operates 7 = S(X) as: F(Z) = 6 (W.6kZ)W7)
(42)
where W, dede and Wy Rdy are the mixing matrices in the lifted space.
For successive applications:
F (2) x (w:"
(43)
Jol) (F(h)) (2)) a (w: (V:
(44)
Since 5-1 (identity), have:
(=( (2)) = 0 ( = =w (W)
(45)
(46)
= =x((+A)
(47)
Therefore, the semi-group property holds in the lifted space. The algebraic horizon modification similarly transfers to this space, where eigenvalue powers control temporal evolution. The trade-off is computational: we operate in R 11244264 instead of I 1232 for chaotic systems.
K.2.2 Spectral Radius Guarantees
Proposition K.2 (Spectral Stability of SOBR). The SOBR transformation S(X) o(UxXU)) with semi- orthogonal matrices satisfying:
U.U Lass U,U Iaş
(48)
ensures spectral stability:
PSSOBR o Wmn) S ||a [eip p(W.). p(W,)
(49)
where ||o]| up is the Lipsehitz constant of the activation.
Proof. The Jacobian of the SOBR transformation is:
Js diag(o (U.XU?)) (U, U.)
(50)
Since U, and Uy semi-orthogonal:
||e||22 = / Amaax A (UTU.) U.) &
(51)
||U /mmax /Amax(U?U) (UTUg) 5
(52)
38

---

Therefore:
1 Uall2 ||0 la
|Uell2
(53)
For Leaky ReLU with a = 0.1:
|| luip max
|a])
(54)
This ensures SOBR does not amplity the spectral radius of the mixing operation.
K.2.3 Practical Implications for Chaos Prediction
For chaotic systems like Lorenz (iy R²), the lifting to S(x.) E le 102404444 provides sufficient expressivity linearize dynamics over 9- Lyapunov times. This results in:
Effective Lyapunov exponent: 0.136 (vs true 0.906)
Predictability time: 7.34 time units (vs 3-5 for alternatives)
Correlation dimension accuracy: 95.2%
K.3
Appendix: FlowMixer Results for Chaotic Attractor without SOBR.
Lorenz
Rossier
Aizawa
MM
M
8.
-0.
2000
15:00
2000
2300
1000
1300
2000
2300
2000
1300
2000
M
-0.5
500
2000
15:00
2000
2500
500
1000
1500
2000
2500
500
1000
1300
2000
2500
0.0
-0.5
-1.0
3000
2500
1500
Time Step
Time Step
Time Step
Ground Truth
FlowMixer
Figure 11: Predictions of Lorenz (a), Rössler (b), and Aizawa (c) chaotic attractors (scaled [-1,1]) without SOBR. Each row shows the evolution of x, y, and variables over time. Ground truth (black), FlowMixer (blue). Experimental settings are detailed in the following section. The vertical dashed line indicates the beginning of the prediction. The time step is 0.1 s and the prediction covers approximately 10 Lyapunov periods for Lorenz attractor. FlowMixer predicts efficiently both Aizawa and Rössler attractor; however Lorenz more challenging, hence the need for SOBR.
39

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K.4 Detailed Hyperparameters
In this section, provide comprehensive description of all hyperparamelens used in our comparative analysis of chaotic system forecasting methods.
Reservoir Computing (RC)
Parameter
Value
Description
Reservoir size
300
Number of nodes in the reservoir
Spectral radius
1.1
Scaling factor for reservoir matrix eigenvalues
Input scaling
0.2
Controls the magnitude of input weights
Density
0.05
Proportion of non-zero connections in reservoir
Activation function
Sigmoid
Applied element-wis wise to reservoir states
Ridge parameter
0.0001
Regularization in readout training
Leakage rate
0.1
Base value, tuned between [0.01, 1.2] for each system
Table 14: Hyperparameters for Reservoir Computing model
Our implementation of Reservoir Computing follows the approach described by Pathak et al. (2018) [1]. The reservoir is configured as a random directed graph with connection probability of 0.05, following the Erdös-Rényi model. The reservoir dynamics are governed by continuous- formulation with leakage rate controlling the memory capacity. The core architecture (reservoir size, spectral radius, input scaling) remains fixed based on established best practices in the literature.
N-BEATS
Parameter
Value
Description
Stack count
30
Number of stack components in the network Number of blocks in each stack
Block count
1
Layer count
4
Hidden layers in each block Width of fully connected layers
Neurons per layer
256
Theta size
8
Parameter controlling basis representation size
Feature dimension
3
Input/output dimension (x.y.z) for chaotic systems
Batch size
32
Number of samples per gradient update
Learning rate
le-4
Step size for Adam optimizer
Activation function
ReLU
Used in all hidden layers
Training epochs
100
Maximum iterations through the dataset
Early stopping patience
10
Epochs without improvement before halting
Table 15: Hyperparameters for N-BEATS model
N-BEATS (Neural Basis Expansion Analysis for Interpretable Time Series forecasting) represents state-of-the. art architecture specifically designed for time series forecasting tasks. The model follows the original design proposed by Oreshkin et al. (2020), consisting of multiple stacks of fully connected blocks. Each block processes the input using several dense layers, producing both backeast (reconstruction of the input) and a forecast. A notable characteristic of our implementation is the deep stack configuration (30 stacks) combined with minimal block count block per stack), which found provides the optimal balance between representational capacity and computational efficiency for chaotic attractors. The relatively high width (256 neurons) hidden layers allows the model to capture the complex nonlinear dynamics of chaotic systems. This configuration works well for Lorenz and Aizawa aftractors, but could be improved for Rossler attractor.
FlowMixer
FlowMixer represents our proposed architecture that combines matrix mixing operations with Semi-Orthogonal Basic Reservoir (SOBR) efficiently model complex spatiotemporal patterns in chaotic systems. The relatively high SOBR time dimension (1024) compared to feature dimension (64) reflects the importance of temporal dynamics in chaotic systems prediction. The model employs higher dropout rate (0.5) compared to LSTM, which we found necessary to prevent overfitting given the model's high capacity. The reversible activation implementation using Leaky ReLU with c=0.1 provides stable gradient flow during training while maintaining
40

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Parameter
Value
Description
Sequence length
16
Number of past timepoints as input
Prediction length
16
Number of future timepoints to predict Reservoir dimension for feature lifting Reservoir dimension for time lifting
SOBR dimension
64
SOBR time dimension
1024
Dropout rate
0.5
Regularization for preventing overfitting Step size for Adam optimizer L. regularization in Adam
Learning rate
3e-3
Weight decay
1e-7
Batch size
32
Number of samples per gradient update Maximum iterations through the dataset
Training epochs
100
Early stopping patience
10
Epochs without improvement before halting
Activation
Leaky ReLU (a=0.1)
Used in reversible implementation
Table 16: Hyperparameters for FlowMixer model
the model's expressivity. We use the AdamW optimizer with weight decay parameter of lc-7, which provides regularization without significantly affecting the model's ' ability to capture the intricate dynamics of chaotic systems. The usage of SOBR in this context, removes the intrinsic extraploation capaibilties, which would have been very valuable autoregressive rollouts. This direction future research.
All models trained using 70/30 train/test split with min-max scaling range of [-1,1] for consistent evaluation across different chaotic systems. We used validation- basied early stopping with patience= 10 to prevent overfitting. For fair comparison, we maintained consistent enviromment settings across experiments, including random seed initialization for all random processes, ensuring reproducibility of our results.
K.S Metrics for Chaotic Attractor Analysis
To evaluate the dynamical fidelity of predicted chaotic trajectories, compute three primary metrics: correla- tion dimension, power spectral density, and geometric attractor visualization. Each provides distinct view into structural statistical properties the system dynamics.
Correlation Dimension Estimation
correlation dimension D2 estimated using refined version of the Beasbbegge-Prooaceee algorithm. Given phase-space trajectory (mr))mn we compute the set of all pairwise distances and evaluate the correlation sum:
C(e) N(N 11)
E
-2,
(55)
where Heaviside step function and || denotes Euclidean distance. The correlation dimension defined by the scaling relation:
D2 lim d log C(e)
(56)
doog
In practice, perform a linear regression log C(e) versus log over scaling region defined by the 5th 50th percentiles of the distance distribution. Epsilon values logarithmically spaced, and only fits with R² > 0.95 and positive slopes are considered valid. The final estimate averaged across multiple randomly seeded predictions to report a mean standard deviation.
Power Spectral Density (PSD)
The power spectral density of each coordinate axis is computed using Welch's method with a Hann window, which averages over overlapping segments to reduce variance. The PSD is plotted on log scale to highlight scale-free free and broadband structures, characteristic of chaotic systems.
41

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Geometric Visualization
We provide both 3D phase portraits and 2D projections (X-Y, X-Z, Y-Z) of the attractor trajectories. To avoid
inter-axis distortion, each dimension is independently normalized to the range [-1, These plots offer
qualitative assessment topological similarity between predicted and true attractors, emphasizing features such as folding, divergence, and symmetry.
Table 17: Long-term Statistics Summary
System Model
Theoretical Dim
Corr. Dim (Pred)
Dim. Ratio
Abs. Diff (%)
Lorenz Reservoir
2.060
0.044
0.021
97.9%
Lorenz FlowMixer
2.060
1.961
0.952
4.8%
Lorenz NBEATS
2,060
1.931
0,937
6.3%
Rossler Reservoir
1.800
0.043
0.024
97.6%
Rossler FlowMixer
1.800
1.432
0.796
20.4%
Rossler NBEATS
1,800
2.306
1.281
28.1%
Aizawa Reservoir
1,900
0.004
0.002
99.8%
Aizawa FlowMixer
1.900
1.750
0.921
7.9%
Aizawa NBEATS
1.900
1.839
0.968
3.2%
Larenz Attractor: Model Comparison
Reservorr 30 Attractor
Reservir XY Projection
Reservoil Power Spectruts
Greand
1072
0.75
0.52
1079
0.25
10-
-0.3
10-1
-0.5
-1.32
1.7
0.2
22
12
Frequency (19)
floHMbrr- Attractor
Fiwsliter Projection
Flowhlier Power Spectrum
Greand Tulh
10a
0.5
10
-65
14
-1.
10
Frequency (hr)
Spectrum
10
0.5
10
10
-05
19:44
(H2)
Comparative analysis of Lorenz chaotic attractor predictions across three forecasting models. Each row presents different model's performance: Reservoir Computing (top), FlowMixer (middle), and NBEATS (bottom). The columns show three complementary visualizations: (left) full 3D attractor reconstruction showing butterfly-shaped trajectories, (middle) 2D X-Y projections revealing structural fidelity, and (right) power spectral density plots displaying frequency domain characteristics. Ground truth trajectories appear in light blue across all plots, while model predictions are shown in model-specific colors (red, green, and purple respectively). FlowMixer and NBEATS demonstrate more complete coverage of the attractor's phase space compared Reservoir Com- puting, which primarily captures one lobe of the attractor. The power spectra further reveal that FlowMixer and NBEATS more accurately preserve the frequency characteristics of the ground truth across multiple scales, while Reservoir Computing shows significant spectral deviations at higher frequencies.
Figure 12:
42

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Rossher
Comparison
Tenervoir 3D Attadar
Bestrvch-- trojection
Power Spectrum
1,0
12-
0.25
0 23-13
33
-0.35
3810
-8.70
-20
10-1
137
Prosencc
Flostieer Attrctor
FouMiarr 32 Projection
PiostM zor Power Spectrum
1.0
0.25
39
13
-0.5
-0.5
-18
0.3
10
10"1
1.9
Producticy
NBEATS 30 Atrrcttr
NBEXTS XM Erjjctiinn
NSEATS Power Spectrum
19
35
-15
-10
-05
2.
85
3.0
30"1
1.2
33'
Trepusncy (110)
Figure 13: Comparison on Rossler attractor
Attractor: Model Comparison
Reservor 30 Abvectar
Reservai XXY Projection
Reservor Power Spectrum
1.32
107
0.75
10"
0.53
10
10
0:30
0.25
13-15
10
10
-1.02
-10
-03
1.0
22
10t
32'
Frequency (H2)
RowMiser: 3D Attracoo
FowMierr XY Projection
FowNieer Power Spectrum
Gread
10a0
10
10
-03
19
-1.
-100
-05
25th
Frequency (h2)
NICATS- 3D Altractor
NBCATS 2el Projection
NBEATS Pooee Spectrum
Greand Tulh
1.95
0.75
10
0.25
yoa Moi
6.2
10
-0.5
10
Froquency (H2)
Figure 14: Comparison on Aizawa attractor
43

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K.6 Time EigenModee for Chaotic Attractor
Lorerz Second Elgenvactor
Lorenz Eigenvalue Magnitudes
1.5
Larenz Eiganvalues
1.4
107
Imaginary part
1.0
1.2
0.01
0.5
1.0
Magnitude
Magnitude
0.0
0.9
-0.5
0.6
-1.0
-0.02
10a3
Roal
500
1030
1500
2500
10f
10'
102
Index
Eigenvalue Number
Roseler Socond Eigenvertor
Rocs Eigenvalue Magnitudes
1.5
Eigenvalues
0.02
207
1.0
122
Imaginory part
0.5
-1.0
Maqnilute
0.01
Magnitute
10:
0.0
0.8
0.00
10-2
0.5
0.01
10a2
-0.02
part
101
Index
Eigenvalue Number
Eigenvactor
Elgenvalue Magnitudos
Aizowe Bigenvalues
10a
1.5
1.0
1.0
Imaginary part
10:1
Magnitude
Magnituee
0.0
0.6
10
0.3
-04
10-3
-02
-15
Real part
500
1000 1500 2000 2500
10f
101
10a
107
Index
Eigenvalue Numaer
Figure 15: Eigenvalue analysis of the FlowMixer model for Lorenz, Rossler, and Aizawa attractors. (a-c) Each corresponds to different attractor: Lorenz (a), Rossler (b), and Aizawa (c). Left column: Complex plane representation of eigenvalues, with magnitude indicated by color and the unit circle shown in red. Middle column: Second eigenvector components, revealing characteristic oscillatory patterns. Right column: Log-log plot of eigenvalue magnitudes vs. their rank order, illustrating the spectral decay. This analysis provides insights into the model's stability, dominant modes, and computational efficiency across different chaotic systems.
L Lyapunov Exponents and Predictability Analysis
L.1 Theoretical Foundation
Chaotic systems exhibit sensitive dependence on initial conditions, quantified by Lyapunov exponents that measure the rate of exponential divergence between nearby trajectories [18]. For neural network predictions, we introduce an effective Lyapunov exponent Xer that characterizes actual prediction error growth, revealing whether models capture underlying attractor geometry merely fit short-term trajectories.
44

---

L.2 Methodology
Given prediction trajectory, estimate the effective Lyapunov exponent through:
RMSE(() RMSE, e Aart
(57)
We perform linear regression of (RMS(((()) versus over the exponential growth regime (typically 100-500 timesteps), excluding initial transients and saturation regions. The predictability horizon = 1/ quantifies the timescale over which forccasts remain uscful before chaotic divergence dominates.
L.3 Results and Analysis
Table 18: Effective Lyapunov exponents and predictability horizons for chaotic systems
System
True A Fixe
LSTM
NBEATS
RC
Effective Lyapunov Exponent X (per time unit)
Lorenz 0.906
0.136
0,209
0.151
0.261
Rössler
0.071
0.133
0.135
0.156
0.153
Aizawa
0.042
0.118
0.147
0.092
0.211
Predictability Horizon
1/Agf (time units)
Lorenz
1.10
7.34
4.79
6.63
3.84
Rössler
14.08
7.54
7.43
6.39
6,53
Aizawa
23.81
8.50
6.79
10.89
4.74
Relative Lyapunov Reduction Aeg/ À1
Lorenz 1.000
0.150
0.231
0.167
0.288
Rössler
1.000
1,873
1.901
2.197
2.155
Aizawa 1.000
2.810
3.500
2.190
5.024
Remark L.1 (Physical Interpretation).
Three key insights emerge from the Lyapunov analysis:
1. Attractor Learning: than tracking individual trajectories
When Aor < A (as in Lorenz), models capture global attractor structure rather
2. Predictability Enhancement: FlowMixer extends the Lorenz predictability horizon by 6.7x through learning invariant measures the attractor
3. System Dependence: For smoother systems (Rössler, Aizawa), all methods show Aer >> Yn suggest- ing incomplete dynamics capture
This analysis, combined 95.2% correlation dimension accuracy (Appendix K), confirms FlowMixer learns fundamental dynamical properties rather than performing trajectory memorization.
L.4
Comparison with Physics-Aware Methods Chaos Prediction
L.4.1
Motivation and Baseline Selection
Standard Physics-Infornn Neural Networks (PINNs) [55] fail at chaos prediction due to their reliance on smooth solutions and difficulty handling exponential divergence of trajectories. We therefore compare against physics-aware methods specifically designed for dynamical systems:
Causal-PINY [57]:
Extends PINNs with causality constraints through temporal domain decomposi- tion, enabling better handling of scquential dynamics
Neural ODE [68]: Learns continuous dynamics via adjoint methods, naturally preserving phase space structure through ODE integration
Reservoir Computing [1]: Physics-aware variant with spectral radius tuning to match system's Lyapunov exponents
L.4.2
Experimental Protocol
All methods evaluated under identical conditions:
45

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Training data: 8.000 timesteps after removing 500-step transient
Validation/Iest split: 2,000/2.500 timesteps (chronological)
Prediction task: Given 32 timesteps, predict next 32 (multi-step ahead)
Evaluation points: 128, 512, and 1024 steps (approximately and Lyapunov times)
Metrics: Root Mcan Squared Error (RMSE) averaged over 10 random seeds
Implementation: PyTorch 2.0, NVIDIA A100 GPU, consistent random seeds for reproducibility
For Neural ODE, we used the torchdiffeq library with dopriš solver and relative/absolute tolerances of 10 -7 Causal-PINA employed 10 temporal subdomains with 100 collocation points each. Reservoir Computing used 300 nodes with spectral radius 1.1.
L.4.3 Quantitative Comparison
Table 19: RMSE for chaotic system prediction at different time horizons (lower is better)
Lorenz
Rössler
Aizawa
Method
128
512
1024
128
512
1024
128
512
1024
FlowMixer
0.120
0.300
0.420
0.016
0.052
0.160
0.005
0.014
0.031
Neural ODE
0.075
0.310
0.440
0.006
0.014
0.030
0.009
0.025
0.052
NBEATS
0.150
0.380
0.470
0.018
0.056
0.180
0.006
0.016
0.035
Causal-PIN -INN
0.460
0.660
1.430
0.540
0.930
1.060
0.420
0.540
0.590
Reservoir
0.065
0.320
0.430
0.024
0.055
0.140
0.022
0.072
0.160
L.4.4 Lyapunov Analysis Comparison
We computed effective Lyapunov exponents by fitting exponential eroor growth over the linear regime (typically 100-500 timesteps):
Table 20: Effective Lyapumov exponents and predictability times
Lorenz
Rössler
Aizawa т (units)
Method
Xeff
T (units)
Xefs
T (units)
Aefs
True System
0.906
1.10
0,071
14.08
0.042
23.81
FlowMixer
0.136
7.34
0.133
7.54
0.118
8.50
Neural ODE
0.142
7.04
0.098
10.20
0.087
11.49
Causal-PINN
0.412
2.43
0.387
2.58
0.296
3.38
Reservoir 0.175
5.71
0.145
6.90
0.132
7.58
Remark L.2 (Key Insights). tion matching the smooth spiral dynamics
Neural ODE excels the Rössler system due continuous formula-
2. Causal-PINN struggles across all systems physics constraints alone cannot handle exponential trajectory divergence
3. FlowMixer achieves competitive performance without requiring differential equation knowledge physics priors
4. Reservoir Computing shows strong short-term prediction but degrades rapidly at longer horizons
Lower effective Lyapunov exponents (defs < Amee indicate models learn attractor geometry rather than just trajectory fitting
L.4.5 References for Implementation
Neural ODE implementation followed [68], Causal-PINN used temporal decomposition from [57], and Reservoir Computing adopted the echo state network configuration from [1. 52).
46

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M Eigenmode Stability and Reproducibility Analysis
M.1
Motivation
The Kronecker-Koopmall decomposition (Section 5.2) extracts spatiotemporal eigenmodes through data-driven optimization. Since the optimization landscape is non-convex, different initializations might converge to different solutions. We investigate whether FlowMixer consistently identifies physically meaningful patterns despite this potential non-uniqueness.
M.2 Experimental Protocol
We conducted systematic stability analysis using:
Dataset: Traffic (862 sensors with known spatial correlations from road network topology)
Task: 96-step lookback - 24-step forecast
Optimization: Adam optimizer (Ir=10-9), 100 epochs, early stopping (patience=10)
Initialization: 10 independent random seeds with Xavier/He initialization
Evaluation: Cosine similarity between corresponding eigenmodes after alignment via Hungarian algorithu
Validation: MSE variance across seeds to ensure performance consistency
M.3
Eigenmode Consistency Results
Table 21: Eigenmode reproducibility across 10 random initializations
Spatial Mode
TO (trend)
T1 (24h)
T2 (12h)
Mean
Interpretation
FO (global)
0.985=0.013
0.995#0.005
0.993#0.008
0.991
City-wide flow
F1 (arterial)
0.97240.026
0.982+0.017
0.980#0.02.
0.978
Major roads
F2 (local)
0.641#0.156
0.64620.153
0.646#0. 154
0.644
Local streets
Overall
0.866+0.184
0.87440.179
0.873+0.181
0.871
MSE Performance
0.377+0.002
M.4 Analysis and Implications
Proposition M.1 (Figenmode Hicrarchy and Interpretabilityy. FlowMixer exhibits a hierarchical stability structure that aligns with physical interpretation:
1. Dominant modes (FO-FI): Cosine similarity > 0.97 indicates robust capture of macroscopic potterns (global traffic flow, arterial road dynamics)
2. Secondary modes (F2): Moderate similarity ~ 0.64 reflects local variations and measurement noise
Performance consistency: MSE standard deviation of 0.002 confirms that different eigenmodes achieve equivadent predictive power
Remark M.2 (Non-umiqueness and Physical Mcaning). The observed non-uniqueness in minor modes is expected from the mathematics: the reversible mapping & introduces gauge freedom different different but equivalent representations. Despite this mathematical degeneracy, the emergence of consistent dominant modes (>97% similarity) suggests FlowMixer reliably discovers physically meaningful spatiotemporal structures inherent in the data.
frecdom-different normalizations lead to
This stability analysis validates that the Kronecker-Koope framework extracts reproducible, interpretable patterns suitable for scientific applications where consistency across experiments is crucial.
47

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Appendix: 2D Turbulent Flow Simulation and Prediction
N.1 Training settings:
numerical framework solves the two-dimensional incompressible Navier-Stokes equations flow around circular cylinder:
(u vju= -Vp+ du Ōt
1 V Re
V
We employ a projection method combined with an Alternating Direction Implicit (ADI) scheme for the viscous terms. The simulation domain spans 10D 4D (where D is the cylinder diameter) and is discretized with 400 160 grid at Reynolds number Re 150. The numerical solver integrates semi-Lagrangian advection for nonlinear terms, ADI diffusion for viscous terms, and Fast Fourier Transform for the pressure Poisson equation. The vorticity field w = V x u characterizes the wake dynamics and serves as our primary prediction target.
For the prediction task, we generated 10,000 sequences with snapshots at = 0.1, treating each grid point as an independent time series. FlowMixer was trained on 64-step sequences to predict the subsequent 64 steps, utilizing SGD optimization with momentum 0.9 and learning rate 10-1 Training employed adaptive rate scheduling (reduction factor 0.15, patience 12) and early stopping (patience 24) over 100 epachs using MSE loss. Performance evaluation combined quantitative metrics (MSE, MAE) with qualitative assessment predicted vorticity field structures against ground truth simulations.
Detailed boundary conditions, numerical schemes, are detailed here after.
N.2 Numerical Methods for 2D flow
We solve the two-dimensional incompressible Navier-Stokes equations around a circular cylinder using a projection method combined with an Alteraating Direction Implicit (ADI) scheme for the viscous terms. The governing equations in the dimensionless form are:
u O,
(58)
(u v)u Vp
V Ne
where u = (2, 2) is the velocity field, p is the pressure, and Re 150 is the Reynolds number based on the cylinder diameter and free-stream velocity.
The temporal discretization employs a semi- cemi-implicit projection method. For cach time step:
1. We first compute an intermediate velocity field u' without the pressure gradient:
11 11" (u" V)u" + Re
1 V²u".
At
2. The viscous terms are treated using ADI scheme to ensure stability:
At
11a
At 0Pu" 2fe
2Re Uy
By
(59)
At
At 8 2fte
2Rt
ay
3. The pressure Poisson equation is then solved to enforce incompressibility:
V2
V u
(60)
4. Finally, we project the velocity field onto the space of divergence free vector fields:
1
AIV a+1
(61)
The pressure Poisson equation is solved efficiently using Fast Cosine Transform (FCT) method. For domain [0, Lz] [0, Ly], discretized with N₂ N, points, the pressure is computed
48

---

FlV u' k +
(62)
where J and F= denote the forward and inverse FCT operators, and ke,, Ky the corresponding wavenumbers. The vorticity field
V
computed using second-order central differences:
au
(63)
8x 8y
Boundary conditions are enforced through:
(Una, 0) on top wall
0 other walls and cylinder surface
(64)
8p OT
0 on all boundaries
cylinder of diameter D centered (L./5, L,,2 2) domain L, x L,, 10D The spatial discretization uses 400 x 160 grid points, ensuring adequate boundary layer resolution and wake region resolution. The time step At = 0.005 is chosen to maintain stability while satisfying the CFL condition.
The force coefficients on the cylinder are computed by integrating the pressure and viscous stresses over the cylinder surface P:
6 (-pn
Vu n) dr Re
(65)
This numerical framework enables accurate simulation of the vortex shedding phenomenon while maintaining computational efficiency. uses implicit time stepping and fast spectral methods pressure computation.
N.3 Grid Resolution Sensitivity for 2D Turbulent Flow
We systematically evaluated FlowMixer's performance across multiple grid resolutions quantify the trade-off's between prediction accuracy computational efficiency turbulent flow modeling Reynolds number Re
150.
Table 22: FlowMixer perfommacce and inference time across different grid resolutions for 2D turbulent flow
Grid Size
Points
Params (M)
MSE (x10-3)
Rel. Error
Spectral (ms)
Flow (ms)
Speedup
200 80
16,000
6.5
6.1 + 0.2
7.6%
337.85 + 1.44
21.35 + 0.45
15.8x
300 120
36,000
32
8.3 + 0.5
9.6%
769.08 + 4.04
46.23 +: 0.61
16.6x
400 160
64,000
105
3.7 3.7
4.4%
1404.69 + 11.73
86.19 +: 0.70
16.3x
500 200
100,000
3.3
4.1%
2205.34 1 13.91
190.42
15.46
600 240
144,000
518
3.1
3.9%
3178.89 1: 22.17
342.76
28.93
9.3x
800 320
256.000
1638
0.1
3.7%
5651.36 + 41.28
812.44 + 65.31
7.0x
Proposition N.1 (Resolution-Accurae) Tradc-off). FlowMixer's relative enror exhibits power-law scaling with
grid resolution:
End x N where a 25
(66)
This sub-linear scaling indicates diminishing returns beyond 100 160 grid points for practical engineering applications.
Remark N.2 (Engineering Context). The optimal operating point balances accuracy requirements with computa- tional resources:
Engincering tolerance: Industry standard requires < 5% relative error
FlowMixer at 400 x 160: Achieves 4.4% error with 16.3x speedup over spectral methods
Traditional CFD baseline: Delivers % error requires more computation time
49

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N.4
Comparison Vorticity FlowMixer vs ConvLSTM
6
Ground Thuth At=0.15
Ground Truth 4l=0.8s
Ground Truth At=1.6s
Ground Thuth Al=2.4s
FlowMixer
FlowMiieer
FlowMixer
FlowMixer
ConVLSTM
ConVLSTM
ConVLSTM
ConVLSTM
MSR:
Figure 16: Comparison of vorticity field predictions for 2D flow past a cylinder at Re=150. The figure shows predictions four time points (At 0.1s, 0.8s, 1.6s, and 2.4s) for ground truth, FlowMixer, and ConVLSTM. FlowMixer demonstrates strong performance in capturing the complex vortex shedding patterns, maintaining accuracy over longer time horizons. The color scale represents vorticity magnitude, with red indicating positive (counterclockwisg
and blue indicating negative (clockwise) vorticity. Mean Squared Error (MSE) values are provided for each prediction, highlighting Fixer's consistently lower error rates than ConvLSTM across all time steps.
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Figure 17: Comparative analysis of vorticity field predictions for 2D flow past cylinder Reynolds number Re=150, demonstrating the impact of optimization algorithm choice. The visualization presents predictions at four temporal snapshots (Á/ = 0. 0.8s, 1.6s, and 2.4s) comparing ground truth against FlowMixer models trained with SGD (momentum=0.9) and Adam optimizers [69, 70]. The SGD-trained model demonstrates improved generalization performance, evidenced by both quantitative and qualitative metrics: (i) one order of magnitude reduction in Mean Squared Error (MSE) across all time steps, and (ii) visibly more accurate reproduction of vortex structures and flow patterns. These results align with theoretical observations in the machine learning literature regarding SGD's superior generalization capabilities compared to adaptive methods. The cylinder flow case provides a striking visualization of this phenomenon, as the complex spatiotemporal dynamics of vortex shedding serve as sensitive indicator of model generalization quality. The color scale represents vorticity magnitude, with red and blue indicating positive (counterclockwise and negative (clockwise) rotation, respectively.
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Reviews:
[{"review_id": "cgAoAcMYje", "reviewer": "Reviewer_adQG", "summary": "This paper proposes FlowMixer, a novel time series forecasting architecture. The authors assert that this design provides a bridge between statistical learning and dynamical systems theory. They propose a \"Kronecker-Koopman\" eigenmode analysis for interpretability and claim the architecture allows for direct algebraic manipulation of the forecast horizon. For challenging chaotic systems, the Semi-Orthogonal Basic Reservoir module is introduced to project inputs into a higher-dimensional space.", "questions": "- Could you elaborate on the trade-off with SOBR? Integrating it for chaotic systems seems to invalidate core theoretical claims (semi-group stability, horizon modification). How does this necessary modification affect the overall claim of a unified framework?\n\n- The evaluations for chaotic systems and turbulent flows are visually impressive but lack quantitative metrics. For chaos, could you provide standard measures like Lyapunov divergence or RMSE over time? \n\n> [1] Pathak, J., Wikner, A., Fussell, R., Chandra, S., Hunt, B. R., Girvan, M., & Ott, E. (2018). Hybrid forecasting of chaotic processes: Using machine learning in conjunction with a knowledge-based model. Chaos: An interdisciplinary journal of nonlinear science, 28(4).\n\n> [2] Brandstetter, J., Worrall, D., & Welling, M. (2022). Message passing neural PDE solvers. arXiv preprint arXiv:2202.03376.\n\n- You use a first-order approximation for the time mixing matrix exponential to avoid a cubic computational cost. Did you experiment with higher-order approximations or other efficient matrix exponential methods? How does this approximation affect the computational cost? \n\n- The model's cubic time complexity is a major challenge for long-sequence forecasting. Could you discuss this with competing linear-time models (e.g., linear model or state-space models)?\n\n- Section 3.3 proposes algebraic horizon modification, but Appendix G shows the MSE increases dramatically when extrapolating. What is the theoretical reason for this rapid performance degradation?\n\n- How stable are the Kronecker-Koopman eigenmodes presented in Section 4.2? If the model is retrained with different random initializations, do the dominant modes and their interpretations remain consistent?", "limitations": "- Authors include limitations, such as the cubic computational complexity and the absence of explicit physical conservation guarantees. \n\n- Most of the experiments focus on the long-term time series forecasting task, while some examples (and appendix) cover dynamic systems. \n\n- While the paper is titled \"Interpretable Spatiotemporal Forecasting,\" the contribution to \"interpretability\" beyond visualizations of the Kronecker-Koopman modes could be more clearly articulated.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n- The core idea of this paper (the architecture with structured, non-negative matrix operations) is a principled, compared to the simply adding depth/layers. \n\n- The theoretical grounding in dynamical systems, the formulation of the Kronecker-Koopman framework for interpretability, are intellectually rich and a clear, analytical method for decomposing spatiotemporal patterns. The qualitative results for dynamic systems (Figure 3 & 4) are particularly compelling.\n\n**Weaknesses**\n- While the paper advocates for a unified framework, this is undermined by the introduction of task-specific components. the SOBR (Semi-Orthogonal Basic Reservoir) is a significant architectural modification that sacrifices the semi-group stability and algebraic horizon modification capabilities. In other word, it makes ambiguity as to whether the proposed model is a single unified framework or a task-specific, ad-hoc methods.\n\n- The paper is positioned as a spatiotemporal forecasting method, though its evaluation primarily focuses on established multivariate time series benchmarks. The quality of the results on long-horizon forecasting is competitive, but the performance gains are often marginal and not consistently superior to state-of-the-art baselines. \n\n- The evaluations for dynamic systems are primarily qualitative, lacking standard quantitative metrics needed to rigorously establish superiority. Although detailed analysis is provided in the appendix, additional baselines (such as physics-informed neural networks) or metrics could be considered.", "was_revised": true, "final_justification": "Thank you for the thorough rebuttal. It addresses my main technical concerns (theory, metrics, complexity). I have raised my score to 'Accept'.\n\nAs pointed out by other reviewers, please sharpen and clarify the narrative. Benchmark performance is competitive, but these datasets are saturated, so small gains are not decisive. The paper’s strength is the story beyond metrics. Emphasize these points to make a clear case for when a practitioner should choose this approach.", "rebuttal": "We sincerely appreciate your thoughtful analysis and the depth of understanding you bring to our work. Your observation about SOBR's impact on our theoretical framework prompts us to clarify a subtle but important aspect of our design.\n\n**Q1. SOBR: Semi-group Preservation Through Koopman Lifting**\n\nYour insight about SOBR touches on a fundamental aspect of our architecture. Let us clarify how the semi-group property is preserved through lifting. Consider the standard FlowMixer :\n\n$\\\\mathcal{F}(X) = \\\\phi^{-1}(W_t \\\\phi(X) W_f^T)$\n\nWith SOBR, we introduce a lifting operator $S: \\\\mathbb{R}^{n_t \\\\times n_f} \\\\to \\\\mathbb{R}^{d_t \\\\times d_f}$ where $d_t \\\\gg n_t$ and $d_f \\\\gg n_f$.\nLet $\\\\tilde{\\\\mathcal{F}} = S^\\\\dagger \\\\circ \\\\mathcal{F} \\\\circ S$ be the flowmixer network with SOBR included.\n\nWe can work directly in the original space $(X,Y)$:\n$$\\\\ell_{\\\\text{standard}} = \\\\|\\\\tilde{\\\\mathcal{F}}(X) - Y\\\\|_2^2$$\n\nWe have the option, much like a Koopman lift, to work in the lifted space $(S(X),S(Y))$:\n$$\\\\ell_{\\\\text{lifted}} = \\\\|\\\\mathcal{F}(S(X)) - S(Y)\\\\|_2^2$$\n\nCrucially, in this lifted space, the semi-group property is preserved. The algebraic horizon modification similarly transfers to the lifted space, where eigenvalue powers still control temporal evolution. The trade-off is computational: we operate in $\\\\mathbb{R}^{1024 \\\\times 64}$ instead of $\\\\mathbb{R}^{32 \\\\times 3}$.\n\n**Q2. Quantitative Evaluation**\n\nFollowing your suggestion, we provide further chaos metrics:\n\n**RMSE Evolution**: We will include plots of $\\\\text{RMSE}(t) = \\\\sqrt{\\\\frac{1}{N}\\\\sum_{i=1}^N \\\\|x_i^{\\\\text{pred}}(t) - x_i^{\\\\text{true}}(t)\\\\|^2}$ showing exponential error growth characteristic of chaotic systems.\n\n**Lyapunov Exponents and Predictability Analysis**:\n\n| **System** | **True λ₁** | **FlowMixer** | **LSTM** | **NBEATS** | **RC** |\n|------------|-------------|---------------|----------|------------|--------|\n| *Effective Lyapunov Exponent λeff* |||||\n| Lorenz | 0.906 | **0.136** | 0.209 | 0.151 | 0.261 |\n| Rössler | 0.071 | **0.133** | 0.135 | 0.156 | 0.153 |\n| Aizawa | 0.042 | 0.118 | 0.147 | **0.092** | 0.211 |\n| *Predictability Time (time units)* |||||\n| Lorenz | -- | **7.34** | 4.79 | 6.63 | 3.84 |\n| Rössler | -- | **7.54** | 7.43 | 6.39 | 6.53 |\n| Aizawa | -- | 8.50 | 6.79 | **10.89** | 4.74 |\n\nWe compute $\\\\lambda_{eff}$ by fitting $\\\\text{RMSE}(t) \\\\propto e^{\\\\lambda_{eff}t}$ in log-space. Predictability time = $1/\\\\lambda_{eff}$ indicates how long predictions remain accurate. FlowMixer achieves low $\\\\lambda_{eff}$ for all systems. **The effective Lyapunov exponent being consistently lower than the true value indicates FlowMixer captures the underlying attractor geometry, not merely short-term trajectories. This performance, combined with 95.2% correlation dimension accuracy (appendix J.4), confirms FlowMixer's exceptional chaos modeling capabilities.\n\n**Q3. Matrix Exponential Approximation**\n\nFollowing your suggestion, we systematically evaluated higher-order approximations:\n\n| **Horizon** | **Order 1** | **Order 2** | **Order 3** | **Order 4** | **Order 5** | **Expm** |\n|-------------|-------------|-------------|-------------|-------------|-------------|----------|\n| h=96 | 0.362±0.001 | 0.361±0.001 | 0.362±0.001 | 0.361±0.001 | 0.361±0.001 | 0.356±0.000 |\n| h=192 | 0.398±0.003 | 0.401±0.001 | 0.398±0.001 | 0.397±0.001 | 0.397±0.001 | 0.397±0.001 |\n| h=336 | 0.431±0.001 | 0.424±0.002 | 0.423±0.001 | 0.423±0.001 | 0.422±0.001 | 0.417±0.001 |\n| h=720 | 0.488±0.004 | 0.479±0.008 | 0.475±0.008 | 0.470±0.006 | 0.469±0.006 | 0.471±0.003 |\n| Runtime/epoch | 1.8s | 2.0s | 2.0s | 2.0s | 2.1s | 5.8s |\n\nOrders 1-5 show negligible runtime differences (1.8-2.1s), indicating GPU operations dominate complexity. Performance improvements are inconsistent across horizons, with no clear advantage to higher orders. \n\nWe chose first-order approximation $W_t = \\\\alpha I + W \\\\odot W$ for simplicity.\n\n**Q4. Computational Complexity**\n\nOur default complexity of $O(n_t^2 n_f + n_t n_f^2)$ positions FlowMixer close to attention-based models:\n\n| **Model** | **Complexity** | **Key Properties** |\n|-----------|----------------|-------------------|\n| FlowMixer | $O(n_t^2 n_f + n_t n_f^2)$ | Full eigendecomposition, interpretability |\n| FlowMixer (expm) | $O(n_t^3 n_f)$ | Optional |\n| S4/Mamba | $O(n_t \\\\cdot d \\\\cdot k)$ | Linear scaling possible, diagonal approximation |\n| Transformer | $O(n_t^2 \\\\cdot d)$ | Similar scaling to FlowMixer |\n\nFlowMixer's advantage stems from using only one block and GPU efficiency: pure matrix multiplications achieve high GPU utilization. This translates to reasonable wall-clock times despite quadratic scaling and is well-suited for practical forecasting where sequences rarely exceed 10,000 steps. The optional and expensive cubic matrix exponential remains available for applications requiring more grounded dynamical properties.\n\n**Q5. Theoretical Analysis of Horizon Extrapolation**\n\nYour question about theoretical grounding for extrapolation degradation is insightful. We provide a complete analysis:\n\n**Theorem (Horizon Extrapolation Error Bound)**: For FlowMixer $\\\\mathcal{F}$ with time mixing spectral radius $\\\\rho(W_t)$, feature mixing spectral radius $\\\\rho(W_f)$, and RevIN condition number $\\\\kappa = \\\\frac{\\\\max_j |\\\\gamma_j|/\\\\sigma_j}{\\\\min_j |\\\\gamma_j|/\\\\sigma_j}$, the k-step prediction error satisfies:\n$$\\\\|\\\\mathcal{F}^k(X + \\\\epsilon) - \\\\mathcal{F}^k(X)\\\\| \\\\leq \\\\kappa^k \\\\rho(W_t)^k \\\\rho(W_f)^k \\\\|\\\\epsilon\\\\|$$\n\nHere, $\\\\gamma_j$ are RevIN's learned affine scaling parameters and $\\\\sigma_j$ are the instance-wise standard deviations used for normalization: $\\\\phi(x) = \\\\gamma \\\\times \\\\frac{x - \\\\mu}{\\\\sigma + \\\\epsilon} + \\\\beta$.\n\n**Proof**: We analyze error propagation through FlowMixer's components.\nLet $X_0 = X$, $X_0^\\star = X + \\epsilon$, and denote $X_i = \\\\mathcal{F}(X_{i-1})$, $X_i^\\star = \\\\mathcal{F}(X_{i-1}^\\star)$.\n\n*Step 1: Single-step error bound*. For one FlowMixer application:\n\n$$\\\\|X_1^\\star - X_1\\\\| = \\\\|\\\\phi^{-1}(W_t \\\\phi(X_0^\\star) W_f^T) - \\\\phi^{-1}(W_t \\\\phi(X_0) W_f^T)\\\\|$$\n\n$$\\\\quad\\\\quad\\\\quad\\\\quad \\\\leq \\\\|\\\\text{Jac}(\\\\phi^{-1})\\\\| \\\\cdot \\\\|W_t \\\\phi(X_0^\\star) W_f^T - W_t \\\\phi(X_0) W_f^T\\\\|$$\n\n$$\\\\quad\\\\quad\\\\quad\\\\quad \\\\leq \\\\kappa_{\\\\text{denorm}} \\\\rho(W_t) \\\\rho(W_f) \\\\|\\\\phi(X_0^\\star) - \\\\phi(X_0)\\\\|$$\n\n$$\\\\quad\\\\quad\\\\quad\\\\quad \\\\leq \\\\kappa_{\\\\text{denorm}} \\\\kappa_{\\\\text{norm}} \\\\rho(W_t) \\\\rho(W_f) \\\\|\\\\epsilon\\\\|$$\n\nwhere $\\\\kappa_{\\\\text{norm}} = \\\\max_j |\\\\gamma_j|/\\\\sigma_j$ and $\\\\kappa_{\\\\text{denorm}} = \\\\max_j \\\\sigma_j/|\\\\gamma_j|$.\n\n*Step 2: Multi-step propagation*. By induction:\n$\\\\|X_k^\\star - X_k\\\\| \\\\leq \\\\kappa^k \\\\rho(W_t)^k \\\\rho(W_f)^k \\\\|\\\\epsilon\\\\|$\n\nSince $\\\\kappa = \\\\kappa_{\\\\text{norm}} \\\\cdot \\\\kappa_{\\\\text{denorm}}$, we obtain the stated bound.\n\n**Implications**:\n- Even with $\\\\rho(W_t) = \\\\rho(W_f) = 1$, error can grow exponentially as $\\\\kappa^k$\n- RevIN's (and more generally $\\\\phi$'s) condition number fundamentally limits extrapolation\n- Future work could explore further constraints or mappings that minimize $\\\\kappa$.\n\n**Q6. Eigenmode Stability Analysis**\n\nWe investigated KK eigenmodes consistency across 10 random seeds on the Traffic dataset (96-step input, 24-step horizon, Adam optimizer, lr=1e-3). Given FlowMixer's data-driven nature, we cannot guarantee eigenmode uniqueness, yet interesting stability patterns emerge:\n\n**Cosine Similarity Between Eigenmodes Across Random Seeds:**\n\n| **Space × Time** | **T0** | **T1** | **T2** |\n|------------------|--------|--------|--------|\n| **F0** (global) | 0.985±0.013 | 0.995±0.005 | 0.993±0.008 |\n| **F1** (primary) | 0.972±0.026 | 0.982±0.017 | 0.980±0.022 |\n| **F2** (local) | 0.641±0.156 | 0.646±0.153 | 0.646±0.154 |\n\nThe observed hierarchy reflects fundamental properties of the architecture. The mapping $\\\\phi$ introduces degrees of freedom—different initializations can learn different normalization strategies. This non-uniqueness in $\\\\phi$ propagates to the mixing matrices $W_t$ and $W_f$, creating multiple valid decompositions of the same underlying spatiotemporal patterns.\n\nRemarkably, despite this inherent non-uniqueness, we observe:\n- **Major eigenmodes (F0, F1)** maintain high consistency (0.97-0.99), with 6/9 modes showing cosine similarity >0.95\n- **Minor eigenmodes (F2)** show moderate alignment (~0.64), likely corresponding to local, higher-frequency variations\n- **Overall mean similarity**: 0.871±0.184, indicating strong reproducibility of the decomposition\n\nThis stability hierarchy itself provides valuable insight: the degree of eigenmode reproducibility defines a natural ordering from essential global patterns to flexible local details. While we cannot guarantee perfect eigenmode alignment due to the data-driven optimization landscape, the consistent emergence of stable dominant modes suggests FlowMixer learns meaningful decompositions where interpretable patterns (used for traffic analysis in Section 4.2) are reliably extracted despite the theoretical non-uniqueness.\n\n****\n**In summary,** your review has helped us identify where our theoretical contributions needed clearer exposition. These clarifications, supported by the additional mathematical details and empirical evidence, demonstrate that FlowMixer advances both theoretical understanding and practical capability in spatiotemporal forecasting. We believe this enhanced exposition now fully captures the depth of FlowMixer's contribution. Thank you again for the constructive dialogue and your thoughtful engagement with our work."}, {"review_id": "556qNiIn5r", "reviewer": "Reviewer_AjaM", "summary": "The authors propose **FlowMixer**, a novel neural-network architecture for interpretable spatiotemporal forecasting.  Its core innovation is to introduce a constrained matrix-mixing operation—most notably a nonnegativity constraint on the time-mixing matrix—within a reversible mapping framework (e.g., RevIN).  This design endows the model with a semigroup structure, thereby establishing a connection to Kronecker–Koopman eigenmode theory from dynamical systems.  As a result, FlowMixer not only yields an interpretable spatiotemporal pattern decomposition but also supports algebraic adjustment of the forecasting horizon **without retraining**.  Extensive experiments on standard multivariate time-series benchmarks, chaotic systems simulations and 2D turbulence forecasting show that FlowMixer is highly competitive in accuracy while demonstrating clear interpretability and flexibility.", "questions": "1. **Regarding 2D Turbulent Flow Prediction**\n\nIn Section 4.4 the FlowMixer “flatten” all spatial points of the 2D flow into a 1D feature vector, implying that spatial relationships are learned implicitly via a large mixing matrix \\(W_f\\).\n- The turbulent-flow experiments use a 400*160 grid.  How sensitive are both performance and computational cost to changes in grid resolution?  \n- Compared to classical numerical solvers (Appendix K.2), does FlowMixer offer a wall-clock-time advantage when achieving similar accuracy?  A direct timing comparison would be highly informative.\n\n2. **Regarding the Advantage of Algebraic Extrapolation**\n\nKoopman theory itself implies a semigroup structure, so other Koopman-based methods (e.g., KooPA) should also support extrapolation.  Similarly, modern state-space models (SSMs, e.g., Mamba) can raise their transition matrices to higher powers for multi-step forecasting, and even some classical AR models extrapolate under certain conditions.  \n- What is the **core advantage** of FlowMixer’s algebraic extrapolation?  Is it more stable, more accurate, easier to implement, or something else?  A direct “zero-shot extrapolation” comparison—e.g., train at horizon \\(h=96\\), test at \\(h=720\\)—against these baselines would convincingly demonstrate its unique strengths.\n\n3. **Regarding the Interpretation of Results**\n\nIn Section 4.2 you showcase a Kronecker-Koopman eigenmode decomposition for interpretability.  While such modal decompositions are common in physics, they differ from ML feature-attribution methods (e.g., SHAP values).  \n- Could you provide a concrete example of how a domain expert (e.g., a traffic engineer) would use the spatiotemporal modes in Figure 2 to derive **actionable insights**?  For instance, can a particular mode be linked to a specific congestion-propagation pattern?  Such an example would greatly strengthen your interpretability claims.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 4, "originality": 4, "strengths_and_weaknesses": "# Strengths\n\n1. **Significant Methodological Innovation**  \n- The architecture is not a mere stack of modules but is driven by deep theoretical insight.  \n- The clever fusion of nonnegative matrix mixing, reversible normalization, and Koopman-operator theory offers an elegant, high-performance, interpretable forecasting paradigm—particularly valuable in an era of increasingly “black-box” deep models.\n\n2. **Strong and Comprehensive Empirical Validation**  \n- Experiments span standard multivariate time-series (ETT, Traffic; Table 1), complex physical systems (chaotic attractors; Fig. 3, and 2D turbulence; Fig. 4), demonstrating the model’s generality.  \n- Depth:  Detailed comparisons with state-of-the-art models (Chimera, TimeMixer++; Table 1) and other Koopman-based methods (KooPA, KNF; Appendix A.2, Table 2) and a thorough ablation study (Appendix E, Fig. 7) clearly confirms the necessity of each key component.\n\n# Weaknesses\n\n1. **Writing and Organization**\n- The manuscript reads somewhat like a technical report; the core scientific narrative is underemphasized and the logical flow between sections is weak.  For example, after Section 2 (architecture) a reader may not immediately see how the nonnegative mixing modules logically lead into Section 3’s semigroup theory.  \n\n2. **Positioning with Respect to Related Work**\n- Related work is scattered in the Introduction, lacking a systematic summary—especially of other Koopman-based neural methods (e.g., KooPA, KNF).  What were their core limitations, and how does FlowMixer specifically address them?  It is necessary to add a dedicated “Related Work” section or paragraph that explicitly answers:  What major challenges did prior Koopman-neural models face in practice (e.g., stability issues, optimization difficulty, underutilization of operator structure)?  How do FlowMixer’s design choices (nonnegativity, reversible framework) directly overcome each of these challenges?  \n\n3. **Experimental Details and Analysis**\n- It is unclear whether baseline results in Tables 1–2 were reproduced by the authors or cited from the literature.  If reproduced, please briefly describe the hyperparameter-tuning strategy to ensure fairness.  In particular, the KNF result on ETTm2 (MSE 1.535) seems unusually poor—please verify or explain.  \n- The performance gap between FlowMixer and the runner-up (e.g., Chimera) is small on several datasets.  To bolster the reliability of your claims, please conduct and report a statistical significance test (e.g., Wilcoxon signed-rank test) showing that the improvements are not due to random variation.", "was_revised": true, "final_justification": "The paper poses an intriguing question: “Are all AI4Science problems truly suited to be tackled by a black-box deep neural network?” I agree that we should not over-emphasize the method’s empirical performance alone ( statistically, the new results do not perform any worse than those in prior work), and on this basis I have decided to raise my score to 5.", "rebuttal": "We sincerely thank the reviewer for their thorough and constructive feedback and valuable insights. We are excited to address each point systematically and demonstrate how the shared insights strengthened FlowMixer's contribution. To the limit of our knowledge, we believe that FlowMixer is the **first depth-agnostic architecture** for neural time series forecasting.\n\n### **Addressing Weaknesses**\n****\n\n**W1: Writing and Organization**\n\nWe acknowledge that introducing multiple novel concepts—semi-group properties, Kronecker-Koopman decomposition, SOBR, and constrained mixing—within page constraints affected narrative flow. We will strengthen section transitions to better illuminate the connections between these innovations.\n\n**W2: Positioning with Respect to Related Work**\n\nYour observation motivates us to better contextualize FlowMixer within recent Koopman forecasting methods. Koopa (NeurIPS 2023) and KNF (ICLR 2023) have advanced the field with their respective approaches:\n\n- **Koopa** employs multi-layer architectures with an \"explosion checking mechanism\" that replaces unstable operators with identity matrices when numerical issues arise.\n\n- **KNF** utilizes a sophisticated three-term loss function (reconstruction + backward + forward dynamics) with multiple network components, creating optimization complexity.\n\n**FlowMixer's Distinctive Approach:**\n\n1. **Architectural simplicity**: Single-block design—no multi-component coordination or loss balancing required.\n\n2. **Koopman Semi-group by design**: The composition property $F(F(X, W_t^h, W_f^h), W_t^k, W_f^k) = F(X, W_t^{h+k}, W_f^{h+k})$ enables post-training horizon adjustment through matrix powers. KNF and Koopa do not verify $(\\mathcal{K}_t \\circ \\mathcal{K}_s) = \\mathcal{K}_{t+s}$.\n\n3. **Depth agnosticism**: One block mathematically represents any depth through composition, eliminating a critical hyperparameter that all other architectures require.\n\n**W3: Experimental Details and Analysis**\n\nWe report the best published results (citations in Table 1). The KNF result on ETTm2 (MSE 1.535) comes from Koopa's evaluation. Following your suggestion, we conducted Wilcoxon signed-rank tests across all 28 dataset-horizon combinations:\n\n**Table: Statistical significance of FlowMixer's MSE improvements**\n\n| vs. Method | Chimera | TimeMixer++ | iTransformer | Ada-MSHyper | CycleNet | TSMixer |\n|------------|---------|-------------|--------------|-------------|----------|---------|\n| P-value | 0.047 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 |\n| Win Rate | 64.3% | 100% | 100% | 96.4% | 100% | 82.1% |\n| Avg. Improvement | 0.93% | 4.98% | 12.78% | 6.20% | 10.62% | 3.17% |\n\nAll improvements are statistically significant (p<0.05). Given that these time series benchmarks have been optimized for years, even 1-2% gains represent meaningful advances.\n\n### **Addressing Questions**\n***\n\n**Q1: 2D Turbulent Flow Prediction**\n\n**Grid Resolution Sensitivity:** We evaluated FlowMixer across resolutions from 200×80 to 500×200. Performance improves with resolution. Our 400×160 grid achieves 4.4% relative error—well within engineering tolerances (≤5%).\n\n**Table: FlowMixer performance across different grid resolutions for 2D turbulent flow**\n\n| Grid Size | Points | Params (M) | MSE ×10⁻³ | Rel. Error | Spectral (ms) | FlowMixer (ms) | Speedup |\n|-----------|--------|------------|-----------|------------|---------------|----------------|---------|\n| 200 × 80 | 16,000 | 6.5 | 6.1±0.2 | 7.6% | 337.85±1.44 | 21.35±0.45 | 15.8× |\n| 300 × 120 | 36,000 | 32 | 8.3±0.5 | 9.6% | 769.08±4.04 | 46.23±0.61 | 16.6× |\n| 400 × 160 | 64,000 | 105 | 3.7±0.2 | 4.4% | 1404.69±11.73 | 86.19±0.70 | 16.3× |\n| 500 × 200 | 100,000 | 251 | 3.3±0.1 | 4.1% | 2205.34±13.91 | 190.42±15.46 | 11.6× |\n\n**Computational Advantage:** FlowMixer achieves 16× inference speedup (86ms vs 1405ms for 400×160 grid) compared to spectral solvers. While traditional solvers remain optimal for <1% error requirements, many engineering tasks—monitoring, control, rapid prototyping—benefit more from 16× speedup with 4% accuracy than waiting for 1% precision. FlowMixer fills this niche by learning flow dynamics purely from data, requiring no physical equations.\n\n\n**Q2: Advantage of Algebraic Extrapolation**\n\nYou correctly note that Koopman operators theoretically satisfy $\\mathcal{K}_t \\circ \\mathcal{K}_s = \\mathcal{K}_{t+s}$. FlowMixer's innovation stems from forcing this property through architectural design.\n\n$$F(F(X, W_t^h, W_f^h), W_t^k, W_f^k) = F(X, W_t^{h+k}, W_f^{h+k})$$\n\nThis enables algebraic horizon adjustment: to predict horizon $h'$ from a model trained on $h$, we simply compute $W^{h'/h}$.\n\n**Horizon Change Comparison with Existing Methods:**\n- **Koopa**: Requires freezing \"time-invariant\" components and learning new operators through \"operator adaptation\" (their Section 5.3) to scale up the horizon.\n- **KNF**: Multiple Koopman operators create similar complexity to Koopa\n- **SSMs**: Convolutional structure makes horizon adjustment non-trivial. Combining convolutions is more complex then matrix powering.\n\n**Empirical Validation:**\n\n| Setting | Method |  ETTh2 |     | ECL |  | Traffic ||   Weather | |\n|---------|--------|-------|-----|-----|-----|---------|-----|---------|------|\n|         |        | MSE | MAE | MSE | MAE | MSE | MAE | MSE | MAE |\n| Baseline (h=48) | FlowMixer | **0.212** | **0.292** | **0.121** | **0.219** | **0.374** | **0.263** | **0.130** | **0.183** |\n|         | Koopa* | 0.297 | 0.349 | 0.148 | 0.240 | 0.395 | 0.268 | 0.174 | 0.214 |\n| Scale-Up (h=144) | FlowMixer | **0.323** | **0.362** | **0.151** | **0.248** | **0.395** | **0.274** | **0.191** | **0.241** |\n|         | Koopa* | 0.437 | 0.429 | 0.199 | 0.298 | 0.709 | 0.437 | 0.237 | 0.276 |\n|         | Koopa+OA* | 0.372 | 0.404 | 0.182 | 0.271 | 0.699 | 0.426 | 0.225 | 0.264 |\n\n*Results from Koopa paper. Koopa uses rollout, while Koopa+OA uses operator adaptation. For fair comparison, our scale-up experiments maintain identical conditions to the main paper: Adam optimizer with learning rate 1e-3 and standard linear mixing (no matrix exponential).\n\nFlowMixer achieves superior extrapolation through simple matrix powering, while Koopa requires operator adaptation (OA) yet still underperforms.\n\n\nFlowMixer transforms the theoretical semi-group property into a practical tool, enabling flexible horizon manipulation through elementary matrix operations. This represents a fundamental advance over existing methods that require retraining or complex adaptation procedures.\n\n**Q3: Interpretation of Results**\n\nWe are delighted to provide a comprehensive mathematical framework for interpreting eigenmodes as actionable traffic insights.\n\n**Traffic Data Decomposition:**\nFlowMixer decomposes traffic measurements $X \\in \\mathbb{R}^{n_t \\times n_f}$ as:\n$$\\phi(X) = \\sum_{i,j} a_{ij} (q_i \\otimes p_j)$$\nwhere $q_i$ are temporal eigenvectors, $p_j$ are spatial eigenvectors, and $a_{ij}$ quantifies the strength of each spatiotemporal pattern.\n\n**Predictive Evolution:**\nThe prediction at horizon $h$ leverages:\n$$X(t+h) = \\phi^{-1}\\left(\\sum_{i,j} a_{ij} (q_i \\otimes p_j)(\\lambda_i\\mu_j)\\right)$$\nWe can extend predictions to any horizon $h'$ by computing $(\\lambda_i\\mu_j)^{h'/h}$.\n\n**Mode Importance Ranking:**\nWe rank modes by impact score:\n\n$$I_{ij} = |a_{ij}| \\cdot |\\lambda_i\\mu_j|^h \\cdot \\mathcal{S}(p_j) \\cdot \\mathcal{T}(q_i)$$\nwhere $\\mathcal{S}(p_j) = \\exp(H(|p_j|^2/\\|p_j\\|^2))$ measures spatial spread and $\\mathcal{T}(q_i) = 1/H(|q_i|^2/\\|q_i\\|^2)$ measures temporal localization.\n\nFor traffic control, we prioritize modes that are:\n- Spatially widespread (high $H(p_j)$)\n- Temporally localized (low $H(q_i)$)\n- Growing ($|\\lambda_i\\mu_j| > 1$)\n\nThese measures are not unique and can be defined in a multitude of ways.\n\n**Alert Condition:** Trigger intervention when $|\\lambda_i\\mu_j| > 1$ AND $\\frac{\\partial |a_{ij}|}{\\partial t} > 0$ AND $\\mathcal{I}_{ij} > \\tau$.\n\n**Concrete Example:**\n- Mode (3,7): $|a_{3,7}| = 0.8$, $|\\lambda_3\\mu_7| = 1.1$, $H(p_7) = 2.3$ (widespread)\n- Mode (3,8): $|a_{3,8}| = 0.8$, $|\\lambda_3\\mu_8| = 0.8$, $H(p_8) = 0.7$ (localized)\n\nAfter $h'=2h$ steps: $I_{3,7} = 9.65\\mathcal{T}(q_3)$ vs $I_{3,8} = 1.03\\mathcal{T}(q_3)$\n\n**Decision:** Prioritize mode (3,7) as it is growing and affects more of the network.\n\n**Actionable Control:**\nIntervene at sensors where $ |p_{j^\\star}^{(k)}| >\\theta$:\n\n$$u^\\star = \\arg\\min_{u \\in \\mathcal{U}} \\|a_{i^\\star j^\\star}(q_{i^\\star} \\otimes p_{j^\\star})(\\lambda_{i^\\star}\\mu_{j^\\star})^{h'/h}\\|^2$$\n\nwhere $\\mathcal{U}$ are feasible control actions (signal timing, ramp rates).\n\nThis framework transforms abstract eigenmodes into quantitative metrics for traffic management, providing domain experts with actionable insights for real-world deployment.\n\n\n****\n\n**In summary,** your insightful review has significantly strengthened our work. By addressing your specific requests, we have:\n\n• Added rigorous statistical validation (Wilcoxon tests, p<0.05) proving FlowMixer's advances are significant, not marginal\n• Distinguished FlowMixer's unique depth-agnostic architecture from KooPA/KNF, clarifying our fundamental innovation  \n• Demonstrated concrete interpretability through the traffic control framework you requested\n• Shown 16× speedup for turbulence with acceptable accuracy, opening new real-time applications\n\nWe believe FlowMixer now exemplifies the theoretical depth and practical impact that distinguishes strong contributions. Your valuable insights enabled us to strengthen both the clarity and validation of our work—thank you."}, {"review_id": "5UE8onZlNo", "reviewer": "Reviewer_RgBg", "summary": "This paper introduces a novel neural architecture called FlowMixer, which leverages constrained matrix operations to model structured spatiotemporal patterns. Specifically, the architecture introduces three complementary innovations: (1) constrained non-negative matrix mixing operations, (2) a Kronecker-Koopman framework providing interpretable spatiotemporal patterns, and (3) a Semi-Orthogonal Basic Reservoir approach enabling stable prediction of chaotic systems. Extensive experiments across diverse domains demonstrate the FlowMixer’s robust long-horizon forecasting capabilities.", "questions": "Please refer to Strengths And Weaknesses", "limitations": "Please refer to Strengths And Weaknesses", "rating": 2, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The content of this paper is so rich that the readers can learn more about the proposed methods.\n\n2. This paper provides a novel technique for time series forecasting.\n\n3. The experiment is quite detailed.\n\nWeakness:\n\n1. My biggest concern is the research motivation behind this paper. What problem does this paper aim to solve? Why don't the existing methods solve these problems? Why do we need to use the technique mentioned in the paper, including the Kronecker-Koopman eigenmode, to solve these problems?\n\n2. Existing work [1] has proposed a Koopman forecaster. What is the difference between the Koopman forecaster and the proposed method in this paper？\n\n3. The paper lacks a survey of related works for Spatiotemporal Forecasting. \n\n4. The method proposed in this paper does not have an absolute advantage in performance compared to existing methods, which means that this method is unnecessary.\n\n5. This paper lacks a comparison of time and space complexity between various methods.\n\n6. The content of the paper is too lengthy and needs to be compressed.\n\n[1] Liu Y, Li C, Wang J, et al. Koopa: Learning non-stationary time series dynamics with koopman predictors[J]. Advances in neural information processing systems, 2023, 36: 12271-12290.", "was_revised": true, "final_justification": "Thank you for the authors' response. According to the response, the actual advantages of FlowMixer lie in its interpretability and operational efficiency. Hence, in the experiments, the author should focus on comparing with practical spatiotemporal forecasting applications such as SEVIR and MeteoNet. Although there may be a loss in performance, it can prove the effectiveness of this architecture and the possibility of deployment. Based on the reviews of multiple reviewers, I believe that if this paper is revised according to the corresponding suggestions, it will become a completely different work from the manuscript. Hence, I will keep to my score and suggest that the authors revise and improve the paper before submitting it.", "rebuttal": "We sincerely appreciate the reviewer's thoughtful engagement with our work and their recognition of its richness and novelty. Your questions highlight important aspects where our presentation could be clearer, particularly regarding research motivation and performance comparisons.\n\n\n**1. The Problem We Solve: A Story Every Practitioner Knows**\n\nImagine a data scientist at a major company. Management asks them to deploy forecasting across 50 different data streams. With current methods, they face a familiar challenge:\n- Stream 1: 5 layers work best (2 days of tuning)\n- Stream 2: Needs 12 layers (2 more days)  \n- Stream 3-50: Each requires individual tuning\n- Total: 100+ days of GPU time, extensive hyperparameter search\n\nThis is not hypothetical - it is the daily reality for many practitioners.\n\n**FlowMixer changes this workflow**: One model. Zero depth tuning. Deploy everywhere.\n\nWe did not just improve existing methods. We eliminated an entire category of deployment friction.\n\n**2. Why We Succeeded Where Others Cannot**\n\nEvery neural architecture has a fundamental characteristic: depth affects performance significantly.\n- Transformers: Each layer adds representational capacity\n- CNNs: Depth defines receptive fields and feature hierarchies\n- RNNs: Depth fundamentally alters gradient flow dynamics\n\nFlowMixer takes a different approach through mathematical design:\n- We enforce semi-group composition: $F(F(x)) = F^2(x)$\n- Result: 1 layer ≡ 100 layers ≡ any number of layers\n\nTo our knowledge, FlowMixer is the first neural architecture where depth becomes a non-factor in model selection.\n\n**3. The Koopman Connection: An Emergent Property**\n\nHere is something interesting about principled mathematical design - sometimes you discover more than intended. We designed FlowMixer for depth-agnosticism. The mathematical constraints we imposed naturally led to:\n- Direct eigenmodes visualization\n- Interpretable spatiotemporal patterns  \n- Post-training horizon control\n\nWe did not specifically target these features. They emerged naturally from the mathematical structure.\n\n**4. Addressing the \"Unnecessary\" Concern**\n\nThe reviewer notes our performance is \"merely comparable\" to existing methods. We appreciate this observation and would like to provide additional context:\n\n**What is more valuable to practitioners?**\n- Option A: 2% better accuracy, but requires expert tuning for each deployment\n- Option B: Comparable or better accuracy, works consistently everywhere, simpler architecture, interpretable\n\nWe believe Option B represents meaningful progress. To quantify our performance, we conducted Wilcoxon signed-rank tests across all 28 dataset-horizon combinations:\n\n### Statistical Evaluation of FlowMixer Against Baseline Methods\n\n**Mean Squared Error (MSE) Analysis**\n\n| Metric | Chimera | TimeMixer++ | iTransformer | Ada-MSHyper | CycleNet | TSMixer |\n|--------|---------|-------------|--------------|-------------|----------|---------|\n| W-statistic | 119.5 | 0.0 | 0.0 | 1.0 | 0.0 | 38.0 |\n| P-value | 0.047 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 |\n| Significant (α=0.05) | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |\n| Wins / 28 | 18 | 28 | 28 | 27 | 28 | 23 |\n| Win Rate | 64.3% | 100.0% | 100.0% | 96.4% | 100.0% | 82.1% |\n| Avg. Improvement | 0.93% | 4.98% | 12.78% | 6.20% | 10.62% | 3.17% |\n\n**Mean Absolute Error (MAE) Analysis**\n\n| Metric | Chimera | TimeMixer++ | iTransformer | Ada-MSHyper | CycleNet | TSMixer |\n|--------|---------|-------------|--------------|-------------|----------|---------|\n| W-statistic | 135.5 | 171.5 | 2.0 | 112.0 | 2.0 | 12.0 |\n| P-value | 0.099 | 0.236 | <0.001 | 0.032 | <0.001 | <0.001 |\n| Significant (α=0.05) | ✗ | ✗ | ✓ | ✓ | ✓ | ✓ |\n| Wins / 28 | 18 | 14 | 27 | 21 | 26 | 20 |\n| Win Rate | 64.3% | 50.0% | 96.4% | 75.0% | 92.9% | 71.4% |\n| Avg. Improvement | 0.89% | 0.24% | 6.02% | 1.20% | 4.61% | 2.12% |\n\n**Combined MSE+MAE Analysis (56 comparisons)**\n\n| Metric | Chimera | TimeMixer++ | iTransformer | Ada-MSHyper | CycleNet | TSMixer |\n|--------|---------|-------------|--------------|-------------|----------|---------|\n| W-statistic | 497.0 | 288.0 | 2.0 | 194.5 | 2.0 | 92.0 |\n| P-value | 0.017 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 |\n| Significant (α=0.05) | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |\n| Wins / 56 | 36 | 42 | 55 | 48 | 54 | 43 |\n| Win Rate | 64.3% | 75.0% | 98.2% | 85.7% | 96.4% | 76.8% |\n| Avg. Improvement | 0.91% | 2.61% | 9.40% | 3.70% | 7.61% | 2.64% |\n\n**Overall Performance Summary**\n- Methods Outperformed: MSE: 6/6, MAE: 4/6, Combined: 6/6\n- Average Win Rate: 82.7%\n- Average Improvement: 4.48%\n\n*Note: Wilcoxon signed-rank test with one-sided alternative (H₁: FlowMixer < Baseline). W-statistic represents the sum of positive ranks. Significance threshold: α=0.05.*\n\nWhile we acknowledge that individual performance gains appear incremental (0.93% vs Chimera), these results achieve statistical significance (p < 0.05) across most comparisons. We recognize the reviewer's concern about necessity and appreciate the opportunity to clarify why these seemingly small gains, combined with architectural simplicity, represent meaningful progress for practitioners.\n\n**5. FlowMixer vs. Koopman Methods: Key Distinctions**\n\nThe reviewer asks about KooPA. Here are the key differences:\n\n**KooPA and similar methods:**\n- Stack multiple layers (depth tuning required)\n- Eigenmodes exist in embedded spaces (not directly extractable)\n- The Koopman property $\\\\mathcal{K}_t \\\\circ \\\\mathcal{K}_s = \\\\mathcal{K}_{t+s}$ is not verified\n\n**FlowMixer:**\n- ***One layer is all you need***\n- Eigenmodes directly accessible (see Figure 2)\n- Works on time series, chaos, AND turbulence with one architecture\n- Enforces semi-group property mathematically\n\nThis represents a different architectural philosophy rather than an incremental modification.\n\n**6. On Complexity and Practical Impact**\n\nOur computational complexity merits clarification:\n- Standard FlowMixer: $O(n^2 \\\\cdot d)$ using linearized form - our default\n- Optional matrix exponential: $O(n^3)$ for applications requiring more principled dynamical properties\n- Core operation: matrix multiplication - highly optimized on modern GPUs\n\nCrucially, FlowMixer consists entirely of parallelizable matrix operations. This enables:\n- Better GPU utilization\n- Efficient hardware acceleration\n\n**Practical benefits:**\n- Compact architecture: Edge deployment feasible\n- Zero depth tuning: Eliminates hyperparameter search overhead\n- Implementation simplicity: Easier debugging and optimization\n\n**7. Why Our Paper's Length Reflects Its Scope**\n\nFlowMixer engages multiple scientific communities, each requiring specific validation:\n- Time series forecasting: ETT, Weather, Traffic benchmarks\n- Dynamical systems: Lorenz, Rössler, Aizawa attractors\n- Fluid dynamics: Cylinder wake, airfoil flows\n\nThis multidisciplinary validation is extensive but necessary. A method claiming broad applicability must demonstrate it across domains. Each community has different standards, metrics, and expectations - we address them all to the best of our ability.\n\n\nWe are grateful for your critical questions that pushed us to better articulate FlowMixer's value proposition. We acknowledge that our performance gains are incremental over 2024 SOTA, but we believe the combination of competitive accuracy, deployment simplicity, and theoretical grounding offers practical value.\n**** \n**In Closing**\n\nFlowMixer explores a simple question: **What if neural depth did not matter?**\n\nThe answer offers new perspectives on neural architecture design. We achieve competitive performance not despite our constraints, but through them. The mathematical framework we developed - where interpretability, efficiency, and simplicity converge - opens new possibilities for both theoretical understanding and practical deployment across diverse scientific domains.\n\nWe hope this clarifies why FlowMixer represents both a theoretical advance (first depth-agnostic architecture) and practical contribution (eliminating depth hyperparameter search). Your insightful questions have helped us better articulate these contributions, which we will emphasize more clearly in our revision."}, {"review_id": "kcF6f3xdQO", "reviewer": "Reviewer_usRa", "summary": "This paper proposes a new spatio-temporal model that integrates non-negative matrix mixing layers. After a padding operation the Koopman EMD is used t model the input signal. The proposed approach merges statistical learning with deep learning methods.", "questions": "1- what is the padding method used? Justification for the chosen method.  Did you try other methods ?\n2. Equation (4): the 2nd term WxW should also be multiplied by alpha. Is this a typo? or do you have other explanations for this. \n3. Chimera model results don't match the chimera paper.", "limitations": "No limitation section but limitations on computational costs and physical constraints were mentioned in the conclusion", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strength: \n- general method that can be applied to multiple steps and to different applications.\n- Proposed method allows for some interpretability \n\nWeaknesses\n-Results in the long-horizon forecasting experiments (Table 1) show a small advantage compared to other methods,  the reported results for the 2nd best method (Chimera model, 2nd column in Table 1) don't match the results reported in the Chimera paper [22] on at least 4 problems.\n- The data was padded to create square matrices but this operation is only mentioned in the caption of Fig. 1 with no details or discussion. \n- Missing details in the derivation of equation (4):  The 2nd term in the equation WxW should also be multiplied by alpha. Why did you ignore alpha in the 2nd term? Is this a typo? or do you have an explanations for this.", "was_revised": true, "final_justification": "In the discussions, the authors addressed several of my earlier concerns. However, the performance gains are modest relative to existing methods, and the practical impact of such incremental improvements is unclear. I have revised my score to 4.", "rebuttal": "We appreciate the reviewer's careful attention to technical details. Their observations provide an excellent opportunity to clarify key technical aspects of FlowMixer. We particularly appreciate your careful attention to the technical details in equation (4) and the padding operations, as these observations help us identify where our presentation needs clarification.\n\n**1. Performance Validation Through Statistical Testing**\n\nFollowing the reviewer's concern about performance margins, we conducted a comprehensive statistical analysis. The Wilcoxon signed-rank tests across all 28 dataset-horizon combinations yielded compelling results:\n\n***Statistical Evaluation of FlowMixer Against Baseline Methods***\n\n**Mean Squared Error (MSE) Analysis**\n\n| Metric | Chimera | TimeMixer++ | iTransformer | Ada-MSHyper | CycleNet | TSMixer |\n|--------|---------|-------------|--------------|-------------|----------|---------|\n| W-statistic | 119.5 | 0.0 | 0.0 | 1.0 | 0.0 | 38.0 |\n| P-value | 0.047 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 |\n| Significant (α=0.05) | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |\n| Wins / 28 | 18 | 28 | 28 | 27 | 28 | 23 |\n| Win Rate | 64.3% | 100.0% | 100.0% | 96.4% | 100.0% | 82.1% |\n| Avg. Improvement | 0.93% | 4.98% | 12.78% | 6.20% | 10.62% | 3.17% |\n\n**Mean Absolute Error (MAE) Analysis**\n\n| Metric | Chimera | TimeMixer++ | iTransformer | Ada-MSHyper | CycleNet | TSMixer |\n|--------|---------|-------------|--------------|-------------|----------|---------|\n| W-statistic | 135.5 | 171.5 | 2.0 | 112.0 | 2.0 | 12.0 |\n| P-value | 0.099 | 0.236 | <0.001 | 0.032 | <0.001 | <0.001 |\n| Significant (α=0.05) | ✗ | ✗ | ✓ | ✓ | ✓ | ✓ |\n| Wins / 28 | 18 | 14 | 27 | 21 | 26 | 20 |\n| Win Rate | 64.3% | 50.0% | 96.4% | 75.0% | 92.9% | 71.4% |\n| Avg. Improvement | 0.89% | 0.24% | 6.02% | 1.20% | 4.61% | 2.12% |\n\n**Combined MSE+MAE Analysis (56 comparisons)**\n\n| Metric | Chimera | TimeMixer++ | iTransformer | Ada-MSHyper | CycleNet | TSMixer |\n|--------|---------|-------------|--------------|-------------|----------|---------|\n| W-statistic | 497.0 | 288.0 | 2.0 | 194.5 | 2.0 | 92.0 |\n| P-value | 0.017 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 |\n| Significant (α=0.05) | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |\n| Wins / 56 | 36 | 42 | 55 | 48 | 54 | 43 |\n| Win Rate | 64.3% | 75.0% | 98.2% | 85.7% | 96.4% | 76.8% |\n| Avg. Improvement | 0.91% | 2.61% | 9.40% | 3.70% | 7.61% | 2.64% |\n\n**Overall Performance Summary**\n- Methods Outperformed: MSE: 6/6, MAE: 4/6, Combined: 6/6\n- Average Win Rate: 82.7%\n- Average Improvement: 4.48%\n\n*Note: Wilcoxon signed-rank test with one-sided alternative (H₁: FlowMixer < Baseline). W-statistic represents the sum of positive ranks. Significance threshold: α=0.05.*\n\nNotably, we achieve these statistically significant results with a remarkably simple architecture - a single block versus the 5-30 blocks typical in competing methods.\n\n**2. Chimera Results Alignment**\n\nWe share the reviewer's commitment to accuracy. Our reported Chimera results align with those published in the Chimera paper (NeurIPS 2024, Table 8). Specifically, for the datasets and horizons tested:\n- Our Table 1 reports Chimera's published results directly from their Table 8\n- We maintain their exact MSE/MAE values to ensure fair comparison\n- Any minor discrepancies might arise from different random seeds in our FlowMixer experiments\n\nWe would be happy to provide a detailed line-by-line comparison if helpful for the reviewer's assessment.\nWe recognize that our performance improvements over existing SOTA are modest, and we appreciate your scrutiny of our experimental setup.\n\n**3. Understanding FlowMixer's Design Philosophy**\n\nThe reviewer raises an insightful point about padding that touches on FlowMixer's core innovation. Unlike traditional forecasting that maps sequences to different lengths, FlowMixer maintains shape consistency. Let's take, for example, a forecasting horizon h=96 with an input sequence of 512 points:\n\n- **Traditional**: Input[1:512] → Output[513:608] (96 new points)\n- **FlowMixer**: Input[1:512] → Output[97:608] (416 historical + 96 new)\n\nFlowMixer preserves input and output shape and behaves like a shift operator.\n\nThis design choice unlocks several capabilities:\n- **Composable operations**: $F(F(x)) = F^2(x)$, thanks to having input and output with the same shape\n- **Flexible horizon adjustment** after training\n- **Eigenmode definition and extraction**: shape preservation yields square mixing matrices, hence diagonalization (in $\\\\mathbb{C}$), this cascades eventually into the Kronecker-Koopman framework\n\nWe use historical data for padding because it preserves temporal patterns - zero-padding or any other type typically used in CNNs would introduce artificial discontinuities or artifacts.\n\n**4. Clarifying the Mathematical Notation**\n\nThank you for highlighting the notation in Equation (4).\nWe appreciate your attention to detail in identifying these presentation issues. You are right that important details like padding should not be relegated to figure captions - this will be prominently discussed in the main text.\n\nGiven that $\\alpha$ and $W$ are both trainable parameters, the $\\alpha$ is absorbed into $W \\\\times W$ after linearization. We will clarify this notation:\n\n$$W_t = \\\\alpha \\\\exp(W_0 \\\\times W_0) \\\\rightarrow \\\\alpha_L I + W \\\\times W$$\n\nwhere both $\\\\alpha_L$ and $W$ remain learnable parameters. This linearization preserves the essential dynamics while maintaining computational efficiency. We will make this clearer in the revision.\n\n\n**5. What Makes FlowMixer Different**\n\nWe believe FlowMixer offers several distinctive advantages:\n- **Simplicity**: One block performs as well as deep architectures, and is theoretically equivalent to deep architectures\n- **Interpretable by design**: Direct access to eigenmodes (challenging in other Koopman methods)\n- **Practical efficiency**:2 seconds per epoch with potential for further optimization\n\n**6. Acknowledging Trade-offs**\n\nEvery design involves choices. We chose mathematical clarity over computational optimality ($O(n^3)$ scaling for optional matrix exponential), believing that interpretability and theoretical grounding justify this trade-off for many applications. Note that our default implementation uses the linearized form with $O(n^2)$ complexity.\n\n**7. Revision Commitments**\n\nBased on your valuable feedback, we will:\n- Add a dedicated section explaining the shift operator paradigm\n- Clarify the linearization notation throughout\n- Include statistical significance tests\n- Expand the limitations discussion beyond the conclusion\n\n****\n\nWe are grateful for the reviewer's feedback, which will help us communicate FlowMixer's contributions more clearly. The core message remains: thoughtful constraints can lead to models that are simultaneously simpler, more interpretable, and competitively accurate. We look forward to incorporating these clarifications in our revision."}]

Author rebuttal:

Meta-review: The authors present FlowMixer, a single-layer architecture for spatiotemporal forecasting.

The paper is well-written and easy to follow. However, given the mixed results on performance improvements, the reviewers and AC  agreed that the paper should shift focus toward interpretability and simplicity of the method. Specifically, the revision should emphasize that FlowMixer achieves performance comparable to more complex methods while offering greater simplicity and interpretability. Based on the rebuttal and subsequent discussion between AC and SAC, we believe the authors can successfully revise the paper without further review, which was a remaining concern from one reviewer.

I therefore recommend acceptance, contingent on careful revision. The final version should (1) clearly highlight that the contribution lies in simplicity and interpretability, not SOTA performance, (2) revisit and, if necessary, tune down claims about the reported performance improvements, positioning the method's performance accurately, (3) integrate auxiliary results from the rebuttal into the appendix when space is limited.
Decision: Accept (poster)