SKOLR: Structured Koopman Operator Linear RNN for Time-Series Forecasting
Authors: Yitian Zhang
Liheng Ma
Antonios Valkanas
Boris N. Oreshkin
Mark Coates
Venue: ICML 2025 poster (2025)
Keywords: Koopman Operator Theory
Linear RNN
Time-Series Forecasting
Primary area: deep_learning->sequential_models_time_series

Abstract: Koopman operator theory provides a framework for nonlinear dynamical system analysis and time-series forecasting by mapping dynamics to a space of real-valued measurement functions, enabling a linear operator representation. Despite the advantage of linearity, the operator is generally infinite-dimensional. Therefore, the objective is to learn measurement functions that yield a tractable finite-dimensional Koopman operator approximation. In this work, we establish a connection between Koopman operator approximation and linear Recurrent Neural Networks (RNNs), which have recently demonstrated remarkable success in sequence modeling. We show that by considering an extended state consisting of lagged observations, we can establish an equivalence between a structured Koopman operator and linear RNN updates. Building on this connection, we present SKOLR, which integrates a learnable spectral decomposition of the input signal with a multilayer perceptron (MLP) as the measurement functions and implements a structured Koopman operator via a highly parallel linear RNN stack. Numerical experiments on various forecasting benchmarks and dynamical systems show that this streamlined, Koopman-theory-based design delivers exceptional performance. Our code is available at: https://github.com/networkslab/SKOLR.

SKOLR: Structured Koopman Operator Linear RNN for Time-Series Forecasting
Yitian Zhang 23 Liheng Ma 123
Antonios Valkanas 123
Boris N. Oreshkin 4
Mark Coates 2
Abstract
1. Introduction
Time-series prediction and analysis of nonlinear dynami- cal systems remain fundamental challenges across various domains. Koopman operator theory (Koopman, 1931) of- fers promising mathematical framework that transforms nonlinear dynamics into linear operations in the space of measurement functions. Practical implementation of this theory faces significant challenges due to the infinite dimen- sionality of the resulting linear operator, necessitating finite dimensional approximations. The dynamic mode decom- position (DMD) and its variants the most widely em- ployed approximations (Rowley et al., 2009; Schmid, 2010; Williams et al., 2015), although alternative techniques have emerged (Bevanda et al., 2021; Khosravi, 2023), including ones employing learnable neural measurement functions (li et al., 2017)
Koopman operator theory provides a framework for nonlinear dynamical system analysis and time- series forecasting by mapping dynamics to a space of real-valued measurement functions, en- abling a linear operator representation. Despite the advantage of linearity, the operator is gener- ally infiite-dimmnsionall. Therefore, the objec- tive to learn measurement functions that yield tractable finit-dimeensioal Koopman opera- tor approximation. In this work, we establish connection between Koopman operator approx- imation and linear Recurrent Neural Networks (RNNs), which have recently demonstrated re- markable success in sequence modeling. We show that by considering an extended state consisting of lagged observations, we can establish an equiv- alence between structured Koopman operator and linear RNN updates. Building on this con- nection, we present SKOLR, which integrates a learnable spectral decomposition of the input signal with a multilayer perceptron (MLP) as the measurement functions and implements a struc- tured Koopman operator via a highly parallel lin- ear RNN stack. Numerical experiments vari- ous forecasting benchmarks and dynamical sys- tems show that this streamlined, Koopman-theon theorr-- based design delivers exceptional performance. Our code is available at: https: / github. com/ http//ttworrsllabbbett
In parallel developments, linear Recurrent Neural Networks (linear RNNs) have emerged powerful architecture in deep learning and sequence modeling (Stolzenburg et al., 2018; Gu & Dao, 2023; Wang et al., 2024b). These models leverage the computational efficiency of linear recurrence while maintaining impressive modeling capabilities.
In this work, consider the task of time-series forecasting and establish both explicit connection and a direct archi- tectural match between Koopman operator approximation and linear RNNs. In particular, we show that by represent- ing the dynamic state using a collection of time-delayed observations, we can establish an equivalence between the application of an extended DMD-style approximation of the Koopman operator and the state update of linear RNN.
Building on this connection, we introduce Structured Koopman Operator Linear RNN (SKOLR) for time-series forecasting. SKOLR implements structured Koopman op- erator through a highly parallel linear RNN stack. Through learnable spectral decomposition of the input signal, the RNN chains jointly attend to different dynamical pat- terns from different representation subspaces, creating a theoretically groundeed yet computationally efficient design that naturally aligns with Koopman principles.
'Department of Electrical and Computer Engineering, McGill University, Montreal, Canada "Mila- Quebce Artificial Intelli- gence Institute, Montreal, Canada 'ILLS International Labora- tory Learning Systems, Montreal, Canada AAmazon Science. This work does not relate the author's position Amazon. Cor- respondence to: Yitian Zhang <yitian.zhang @ mail. mail.meeill..a>. Liheng Ma <liheng.ma ma @ meiill.e
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
Through extensive experiments on various forecasting benchmarks and dynamical systems, we demonstrate that

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SKOLR: Structured Koopman Operator Linear RNN
this streamlined, toopmman-hhooy-bbsees design delivers ex- ceptional performance, while maintaining the simplicity of the linear RNN and its outstanding parameter efficiency.
and input space V C Rdv, For any sequence (v=) E V. the linear RNN defines a discrete-time dynamical system through the hidden state transition equation:
2. Preliminary
he = Wh:-1++ + +Uv +
(4)
This section provides foundational background for the pro- posed forecasting methodology. We define the discrete-time dynamical systems used model observed time series and then introduce the Koopman operator.
where W E RRd dd is the hidden state transition matrix, U e Rd xdu is the weight matrix applied to the input, and b e Rd is the bias vector. The evolution of hidden states h& H is uniquely determined by this linear map.
Definition 2.1 (Discrete-time Dynamical Systems). We con- sider the (autonomous) discrete-time time dynamical system:
In order to prepare the connection to our proposed Koopman operator learning strategy and forecasting method, let us introduce Va := (yyx) and define g(yx) == Uv)(ya) + b for a suitable function sb. Then, if gyyk==) = 0 for 8 > L, we can unroll Eq. 4 to the following form:
F(x:):
x
where X2 E M denotes the system state at time k E Z+; and : -M .M represents the underlying dynamics map- ping the state forward in time. We assume Euclidean state space M C C although it can be more generally defined on an n-dimensional manifold (Bevanda et al., 2021).
h2 = g(yx)))+))))))))))(())) + Wyy(yk-)).
(5)
The Koopman operator framework enables globally linear representations of nonlinear systems by applying a linear operator to measurement functions (observables) g of the state Xg. The following theorem formalizes key properties of the Koopman operator.
Here W indicates s applications of W.
3. Methodology
3.1. Problem Statement
Theorem 2.2 (Koopman Operator Theorem (Koopman, 1931; Brunton et al., 2022)). Considering real-valued mea- -M- + R, the
Let us denote L steps of the trajectory of a discrete-time dynamical system as Xi...., X We focus on the setting where X2 E X C RCC We do not directly observe Xt, but instead observe yı h((xs) for some unknown function h.
surement functions ka. observables) Koopman operator K. : F + F is an infinite-dimensinn linear operator on the space of all possible measurement functions F, which is an infinite-dime dimensional Hilbert space, satisfying:
We have available a set of training data consisting of mul- tiple sequences of length I, + T. The inference task is to forecast the values y yL+1..... LL+T given observations of the first values of a sequence yI....,
Kog= g F
where is the composition operator.
In other words.
3.2. Strategy: Directly observable systems
K(g(xx)) = ((F(xx)) = g(xx+1).
(3)
We first consider the setting where we directly observe the state, i.e., h is the identity, and ys h(xx) = Xx- Assume that the dynamical system be captured by X&+1 F(x.). Then an appropriate forecasting approach is to learn a finite dimensional approximation to the Koopman operator associated with F, and then propagate it forward in time to construct the forecast. In this section, for nota- tional simplicity, we describe a setting where learning is based on a single observation
This is true for any measurement function g and for any state Xg. 1
While this facilitates analysis via linear maps, Koopman op- erator is generally infiniteddimensional, acting on Hilbert space of functions. For practical learning and inference, we seek effective finite-dimensional approximations. In this paper, we construct these approximations efficiently by leveraging connection to linear recurrent neural networks (RNNs). For clarity, we now define linear RNN.
(which, for now, Xo..... xL). The extension learning using multiple observed series is straightforward.
Let us introduce measurement functions 92, Bn H. where H is Hilbert space containing real-valued functions defined on X. Denoting L(H) as the space of bounded linear operators T : H + H. Khosravi (2023) formulates the learning of the Koopman operator as a minimization
Definition 2.3 (Linear Recurrent Neural Network (Stolzen- burg et al., 2018)). Consider hidden state space H C dd.
Koopmman operators can also be defined on continuous-time dynamical system, which we disregard here for now
2

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SKOLR: Structured Koopman Operator Linear RNN
task with Tikhonov-regularized empirical loss:
make it more robust. In particular, we enforce structure
M that allows to write:
L
ng Xxt 1aa-(ke (Kgg()((xx=))) + + (6)
g(xx) Mg = glyk-1)+ W g(yk-)))) (8)
min KEC(F)
With this structure, we see that M is a blockwise diagonal matrix, where each block is a power of a learnable matrix W. Moreover, by comparing Eq. with Eq. 5, we see that this structure, which represents the dynamic state using a collection of time-delayed observations (Arbabi & Mezic, 2017), can be implemented as a linear RNN.
Although this optimization is over an dnfinit-ddimensions
space of linear operators, Khosravi (2023) demonstrates that if the measurement functions satisfy certain conditions, then there is a unique solution K, which can be derived by solving imite-dimensional optimization problem.
A special case occurs when K is invariant in the sub- space spanned by the measurement functions, G span(g...... In, J. i.c. K E C(g). In this setting, we can follow the approach of the Extended Dynamic Mode Decomposition (EDMD) method (Li et al., 2017). which approximates the Koopman operator by a finite-dimensional linear map U : G + G. We can represent the action of U on gi using matrix M R*,x",, Because the di- mension of G is finite, can identify an M such that Ug: = [M The matrix M can then be estimated via the following minimization:
SKOLR
Building on analysis of Koopman operator approxi- mation and the connection to the linear RNN, we present SKOLR, which integrates a learnable spectral decomposi- tion of the input signal with a multilayer perceptron (MLP) for the measurement functions.
Inspired by multiresolution DMD (Kutz al., 2016), instead of learning a single linear RNN acting on high dimensional space, we propose to split the space into multiple subspaces, resulting in learning a structured Koopman operator via highly parallel linear RNN stack. This structure also improves the parameter efficiency, as shown in Fig.
min P&M - Y M
where Pa = Di- and Y = (xx)) (x))) When applying EDMD, we assume that PG is full rank so that a unique minimizer can be identified via the Moore- Penrose pseudoinverse.
k=
Encoder Let the input sequence be Y Y V2, yzl, where yı E R.P, with heing the dimension of the observa- tion. The encoder performs learnable frequency decomposi- tion via reconstruction of the soft-gated frequency spectrum via Fast Fourier Transform (FFT) and Inverse FFT (IFFT):
This learning task can be made more flexible by allowing for learning of the measurement functions gr. Li et al. (2017) propose method that incorporates neural network based learning of the measurement functions and an Ln regularizer to promote sparsity.
S FFT(Y),
S, = S Sigmoid(w,,..
(9)
Y, IIFT(S.))
3.3. Unobserved states
The reconstructed signals (Y,)-2 form N parallel branches, with each branch representing a frequency-basee subspace, while [w,]] contain learnable parameters for frequency selection.
In many settings, we do not observe the state of the system X directly. Instead we observe yk = h(xx)) for some unknown observation function ht-). In this setting, Yk may not provide sufficient information in isolation to recover the state Xx- We can instead construct state representation by considering the past L measurements, i.e., we define Xx = =yy y&] With this representation, we can model the dynamics as X = 11x and target learning of the Koopman operator K associated with F. Note that this permits us to perform prediction, because we are interested in predicting y which can be recovered from Ra+1-
N
For each branch we parameterize the measurement func- tions using non-linear feed-forward network:
Zn, = FFN (yk), for k = L,..,,
(10)
where FFN: : R. + RDD. We utilize multiple layer percep- trons (MLPs) for simplicity, generally with only one layer or two. Other FFNs like wiGLU (Shazeer, 2020) are also applicable. After this encoding is complete, we have con- structed the 2, g(yxs) (and hence the g(xx)) that appear in Eq. 8 for A: = 1,..., L. By structuring the architecture into multiple branches and incorporating both frequency- domain filtering and time-domain encoding, we enhance flexibility in learning suitable measurement functions for diverse time-series patterns.
Under the same assumptions of invariance of the Koopman operator with respect to G, we can adopt the same approach as outlined above, learning a matrix M. Given the structure of the constructed Xs, we are motivated to impose further structure on the Koopman operator matrix, with the goal of introducing an inductive bias that can facilitate learning and
3

---

SKOLR: Structured Koopman Operator Linear RNN
Maa E Rixt
S141447 E RixT
Encoder
Σ L Myj thrn
Σ E st tleal
Σ Eij 8h)
(l)((2)
Bli((r) (1,)(xg))
(1,(
(1,2(x))
81, (1,)(a))
...
(12(1)))
Decoder
Rx
Structured Koopman
Operator Approximation
Mi
M
Mi
M2
M,
hr.1
b2.1
h
B
hg+2.1
By4T.11
M
(HL-T) (T
112
M
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Mr
M:
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11.11.2
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B1+T22
M2
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DEALN
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Figure 1: Architecture of SKOLR (Structured Koopman Operator Linear RNN) The input time series goes through an encoder with learnable frequency decomposition and MLP that models the measurement functions. With the branch decomposition, the highly parallel linear RNN chains jointly attend to different dynamical patterns from different represen- tation subspaces. Finally, a decoder reconstructs predictions by parameterizing the inverse measurement functions. This structured approach maintains computational efficiency while naturally aligning with Koopman principles.
RNN Stack Given the collection for each branch and time step k: Z, (z.,,....; AL,n E RDx take it as input to linear RNN and introduce learnable branch- specific weight matrices W, for each branch.
For prediction of length we recursively apply the operator to predict the Koopman space for future steps per branch:
H H M. M h
(14)
h ha+ W, h + Zk,n
(11)
Decoder For reconstruction, use mirrored feed- forward networks parameterize the inverse measurement functions g-1 The decoder processes the hidden states
Each branch weight matrix W, n defines a matrix Mn, discussed above, which specifies finite-dimensional ap- proximation to a Koopman operator for X for the learned measurement functions on that branch.
ýk.. FFN (b),.)
(15)
Together, the branch matrices M, form structured finite- dimensional Koopman operator approximation K with block diagonal structure:
where FFN RR. R...
The decoder combines predictions from all branches gen- erate the final prediction "14+1,4+7) The model trained end-to-end using the loss function:
M
0
0
0
M
0
K
(12)
M
L = =119p41114 - VILLLLTTIE
(16)
By imposing this structure and using a stack of linear RNNs, we can learn local approximations to the evolution dynamics of different observables. For each branch
The structured approach, induced by both the linear RNN and the branch decomposition, enables efficient parallel pro- cessing and reduces the parameter count. Since all architec- tural components are very simple (basic sigmoid frequency gating, one- or two-layer MLPs for encoding/decoding, lin- car RNN), the architecture is very fast to train and has low memory cost, as we illustrate the experiments section.
L-1
H. [Zi,n Z2,n + MI ZL +
M Zenl
2=0
(13)

---

SKOLR: Structured Koopman Operator Linear RNN
Table I:
Prediction results on benchmark datasets, L = 2T' and I' E {48, 96, 144, 192} (ILI: T E {24, 36, 48, 60)). Best results and second best results are highlighted in red and blue respectively.
Models
T
SKOLR
Koopa MSE
iTranforrmrr MSE
PatchTST MSE
TetexNet
Dlineur MAE
MICN
KNF
Autoformer
MSE
MAE
MAE
MAE
MAE
MSE
MAB
MSE
MSE
MAE
MSE
MAE
MSE
MAE
48
0.137
0.229
0.130
0.234
0.133
0.225
0.147
0.246
0.149
0.254
0.158
0.241
0.156
0.271
0.175
0.265
0.164
0.272
96
0.132
0.225
0.136
0.236
0.134
0.230
0.143
0.241
0.170
0.275
0.153
0.245
0.165
0.277
0.198
0.284
0.182
0.289
ECL
144
0.143
0.236
0.149
0.247
0.145
0.240
0.145
0.241
0.183
0.287
0.152
0.245
0.163
0.274
0.204
0.297
0.210
0.315
192
0.149
0.244
0.156
0.254
0.154
0.249
0.147
0.240
0.189
0.291
0.153
0.246
0.171
0.284
0.245
0.321
0.221
0.324
48
0.400
0,258
0.415
0.274
0.369
0.256
0.426
0.286
0.567
0.306
0.488
0.352
0.496
0.301
0.621
0.382
0.640
0.361
Traffic
96
0.368
0.248
0.401
0.275
0.388
0.270
0.413
0.283
0.357
0.485
0.336
0.511
0.312
0.645
0.376
0.668
0.367
144
0.375
0.255
0.397
0.276
0.375
0.267
0,405
0.278
0.603
0.322
0.452
0.317
0.498
0.309
0.683
0,402
0.681
0.379
192
0.377
0.256
0,403
0.284
0.373
0.267
0.404
0.277
0.604
0.321
0.438
0.309
0.494
0.312
0.699
0.405
0.692
0.385
48
0.131
0.170
0.126
0.168
0.136
0.174
0,140
0.179
0.138
0.191
0.156
0.198
0.157
0.217
0.201
0.288
0.185
0.240
Weather
96
0.154
0.202
0.154
0.205
0.169
0.216
0.160
0.206
0.231
1.186
0.229
0.187
0.250
0.295
0.308
0.230
0.279
144
0.172
0.220
0.172
0.225
0.192
0.242
0,174
0.221
0.190
0,244
0.199
0.244
0.197
0.257
0.394
0.401
0.268
0.308
192
0.193
0.241
0.193
0.241
0.204
0.251
0.195
0.243
0.212
0.265
0.217
0.261
0.214
0.270
0.462
0.437
0.325
0.347
48
0.280
0.330
0,283
0.333
0,313
0.356
0,286
0.336
0.308
0.354
0.322
0.355
0,294
0.353
1.026
0.792
0.592
0.419
ETTml
0.289
0.340
0.294
0.345
0.302
0.353
0.299
0.346
0.329
0,370
0,309
0.346
0.306
0.364
0.957
0.782
0.493
0.469
144
0.319
0
0.322
0,366
0,331
0.374
0.325
0.366
0.358
0.387
0.327
0.359
0,342
0.390
0.921
0.760
0.735
0.569
0.328
0.373
0.337
0.378
0.343
0.381
0.343
0.375
0.462
0.441
0.337
0.365
0.386
0.415
0.896
0.731
0.592
0.506
48
0.134
0.228
0.134
0.226
0.139
0.234
0.135
0.231
0.142
0.234
0.144
0,240
0.131
0.238
0.621
0.623
0.191
0.280
ETTm2
96
0.171
0,255
0.171
0.254
0.177
0.268
0.171
0.255
0.187
0.269
0.172
0.256
0.197
0.295
1.535
1.022
0.241
0.311
144
0.209
0.283
0.206
0,280
0,216
0.296
0.205
0.282
0.216
0,291
0.200
0.276
0.210
0.297
1.337
0.876
0,300
0.352
0.304
0.226
0.298
0.237
0.310
0.221
0.294
0.313
0.290
0.248
0.328
1.355
0.908
0.324
0.370
48
0.333
0,373
0
0.377
0,342
0.380
0.337
0.375
0.365
0.399
0.343
0.371
0.375
0,406
0,876
0.709
0.442
0.438
ETThl
96
0.371
0,398
0.371
0.405
0.393
0.412
0.372
0.393
0.411
0.430
0.379
0.393
0.406
0.429
0.975
0.744
0.634
0.523
144
0.405
0.417
0.405
0.418
0.425
0.430
0.394
0.412
0.442
0.447
0.393
0.403
0.437
0.448
0.801
0.662
0.522
0.491
192
0.422
0.432
0.416
0.429
0,456
0.454
0.416
0.439
0.469
0.470
0.407
0.416
0.518
0.496
0.941
0.744
0.525
0.501
0.238
0.306
0,226
0,300
0.243
0.313
0.223
0.297
0.241
0.226
0.305
0,260
0.336
0.385
0.376
0.355
0.380
ETT22
96
0.299
0,352
0.297
0.349
0,306
0,358
0,300
0.353
0.325
0.376
0.294
0
0,343
0.393
0.433
0.446
0.427
0.452
144
0.335
0.377
0.333
0,381
0,347
0.385
0.346
0.390
0.374
0.408
0.354
0.397
0.374
0.411
0,441
0.456
0.457
0.461
192
0.365
0.397
0.356
0.393
0.375
0.403
0,383
0.406
0.394
0.434
0.385
0.418
0.455
0,464
0,528
0.503
0.503
0.491
1,556
0.760
1.621
0,800
1.763
0.843
2.063
2.464
1.039
2.624
1.118
4.380
1,558
3.722
1.432
2,831
1.085
ILI
36
1.462
0.728
1. 1000
0.855
2.067
0.919
2.178
0.943
2.388
1.007
2.693
1.156
3,314
1.313
3.941
1.448
2:801
1.088
48
1.537
0.798
1.768
0,903
1.667
0.879
1,916
0.896
2.370
1.040
2.852
1.229
2.457
1.085
3,287
1.377
2.322
1.006
60
2.187
0.995
1,743
0.891
2,011
1,000
1,981
0.917
2.193
1.003
2.554
1.144
2.379
1.040
2.974
1.301
2.470
1.061
Rank
7
3
2
|
0
0
0
|
4. Experiments
We select these representative baselines for their established performance and public implementations.
4.1. Benchmarking SKOLR
Following Koopa (Liu et al., 2023), we set the lookback window length L = 2T for prediction horizon T E {48, 96, 144, 192} for all datasets, except ILI, for which we use T E {24, 36, 48, 603. This setting leverages more historical data for longer forecasting horizons. We report baseline results from Liu et al. (2023) except for iTrans- former; we reproduce iTransformer results with L = 27 using the officially released code. Performance is measured using Mean Squared Error (MSH) and Mean Absolute Error (MAE). Appendix A.2 provides implementation details.
1. DATASETS
We evaluate SKOLR on widely-used public benchmark datasets. For long-term forecasting, we use Weather, Traf- fic, Electricity, ILI and four ETT datasets (ETTh1, ETTh2, ETTml, ETTm2). We assess short-term performance on M4 dataset (Makridakis et al., 2020), which includes six subsets of periodically recorded univariate marketing data. For more information about the datasets see Appendix A.1.
4.1.2. BASELINES AND EXPERIMENTAL SETTINGS
4.1.3. RESULTS AND ANALYSIS
We compare against sfateoof-hh-aa deep forecasting mod- els. The comparison includes mansfoommr-bbseee models: Autoformer (Wu et al., 2021), PatchTST (Nie et al., 2023), iTransformer (Liu et al., 2024), TCN-based models: Times- Net (Wu et al., 2023), MICN (Wang et al., 2023a), linear model: DLinear (Zeng et al., 2023), and Koopman- based models: KNF (Wang et al., 2023b), Koopa (Liu et al., 2023).
Table 1 reports the experimental results for eight bench- marks. The performance is measured by MSE and MAE; the best and second-best results for each case (dataset, hori- zon, and metric) are highlighted in bold and underlined, respectively. The results are the average of 3 trials.
We rank the algorithms in Table 1 based on their MSE
5

---

SKOLR: Structured Koopman Operator Linear RNN
SKOLR
400
FFT
Signal
Branch 2 Branch 1 Input
KooPA
200
PatchTST
iTrnnffrmer
400
Input
True
Pred
DLinear
TimesNet
MICN
400
Autoformer
200
KNF
0
20
250
500
750
1000
1250
1500
5 Rank
Frequency
Steps
Figure 3: Analysis of SKOLR's branch-wise behavior: (a) frequency decomposition and (b) prediction performance. We observe that different branches focus on different fre- quency components.
Figure 2: Boxplot for ranks of the algorithms (based on their MSE) across seven datasets and four prediction horizons. The medians and means of the ranks are shown by the vertical lines and the black triangles respectively; whiskers extend to the minimum and maximum ranks.
Table demonstrates SKOLR's effectiveness across differ- ent dynamical systems. For periodic systems like Pendulum, SKOLR achieves substantial improvements, indicating su- perior capture of oscillatory patterns. In chaotic systems like Lorenz '63, SKOLR shows better stability with 10.9% reduction MSE, suggesting robust handling of sensitive dependence initial conditions. The model demonstrates particularly strong performance on mixed dynamics: Lotka- Volterra and Dufting oscillator. These results validate that SKOLR's structured operator design effectively captures both periodic motions and complex nonlinear dynamics.
Table 2:
Model evaluation results (MSE/MAE) on non- linear dynamical systems (NLDS)
SKOLR
KooPA
Dataset
MSE
MAE
MSE
MAE
Pendulum
0.0001
0.0083
0.0039
0.0470
Duffing
0.0047
0.0518
0.0365
0.1479
Lotka-Volterra Volterra
0.0018
0.0354
0.0178
0.1050
Lorenz '63
0.9740
0.7941
1.0937
0,8325
Fig. 3 demonstrates SKOLR's ' multi-scal decomposition strategy. The FFT analysis reveals how different branches place more emphasis on some frequency bands. This nat- ural frequency partitioning emerges from our structured Koopman design, enabling each branch to focus on spe- cific temporal scales. The prediction visualization illustrates the complementary nature of these branches, where their combined forecasts reconstruct complex dynamics through principled superposition of simpler, frequency-specifu pre- dictions. More analysis can be found in Appendix D.3.
and order them based their average rank across eight datasets and four prediction horizons. Figure 2 shows the relative ranks. We observe that SKOLR achieves SOTA performance, with the best average rank across all settings.
The model shows strength in capturing complex patterns in the Weather dataset, matching Koopa's performance while surpassing other transformers, indicating effective handling of meteorological dynamics. For the ILI dataset, which features highly nonlinear epidemic patterns, SKOLR outper- forms the baseline methods, with significant error reduction for the shorter horizons.
4.3. Analysis and Ablation Study
4.3. 1. ANALYSIS: STRUCTURED KOOPMAN OPERATOR
While SKOLR demonstrates strong performance long- term forecasting, we also evaluate its effectiveness on short- term predictions with M4 dataset. Results in Appendix B. show consistent improvements over both transformer-bas based forecasting methods and Koopman-based alternatives across different time scales.
We analyze the impact of branch configurations through two controlled experiments: (1) Fixed parameter count scenario, where total parameters remain constant (~1..MM across con- figurations while varying the learnable frequency decom- position w, with N branches and lookback window L; (2) Fixed dimension scenario: We maintain constant Koopman operator approximation dimension dim(K) 512 while varying branch number N. As N increases, each branch's 's dimension / decreases proportionally (D 512/N), lead- ing to reduced parameter count.
4.2. State Prediction for Non-Linear Systems
Koopman operator-based approaches have gained attention for their ability to perform system identification in fully data-driven manner. To evaluate SKOLR's performance in this context, we conducted a series of experiments on non- linear dynamical systems (NLDS) (details in Appendix E).
We conduct experiments on the ETTml dataset. As we can see in Table and Fig. maintaining similar parameter
6

---

SKOLR: Structured Koopman Operator Linear RNN
MSE with Different Branch Numbers
Table 4: Performance comparison with dim(K) 512 and varying branch numbers N on ETTml. Parameter counts shown for horizon T = 192.
N
0.290
16
MSE
0.285
Config.
Params
MSF for Different T
(D, N)
(M)
96
144
192
0.280
(512, 1)
1.71
0.284
0.292
0.326
0.330
(256, 2)
0,92
0.280
0.294
0.318
0.334
T-48
T-96
(128, 4)
0.53
0.280
0.293
0.318
0.335
(64, 8)
0.34
0.283
0.297
0.316
0.329
Figure 4: MSE comparison on ETTml dataset across dif- ferent branch configurations and prediction horizons. Bars show MSE values for each configuration. All configura- tions maintain similar parameter counts (~1.6M). Increas- ing branch number improves performance.
(32, 16)
0.25
0.282
0.292
0.313
0.328
Table Ablation Study on frequency decomposition
SKOLR (learn)
SKOLR(W 1)
Dataset
T
Table 3:
Performance comparison with similar parameter counts on ETTml
MSE
MAE
MSE
MAE
48
0.137
0.229
0.150
0.239
96
0.132
0.225
0.134
0.227
ECL
Config.
MSE for Different T
144
0.143
0.236
0.144
0.237
192
0.149
0.244
0.150
0.244
(D, N)
96
144
192
48
0.131
0.170
0.134
0.173
(512, 1)
0,284
0.292
0.326
0.330
96
0.154
0.202
0.158
0.203
Weather
(256, 4)
0.282
0.291
0.317
0.341
144
0.172
0.220
0.175
0.221
(128, 16)
0.281
0.287
0.323
0.329
192
0.193
0.241
0.194
0.242
48
0.333
0.373
0.329
0.371
96
0.371
0.398
0.375
0.400
counts (~1.66M and increasing branch numbers from N -1 to N = 16 improves performance for most horizons.
ETThl
144
0.405
0.417
0.407
0.419
192
0.422
0.432
0.429
0.433
More significantly, when keeping dim(K) (K) =512 (Table 4), models with more branches maintain strong performance despite substantial parameter reduction. Notably, the con- figuration with D 32,N N 16 achieves comparable performance the 1-branch model while using only 0.25M parameters (85% reduction). This demonstrates that struc- tured decomposition through multiple branches enables sig- nificantly more efficient parameter utilization while main- taining or improving forecasting accuracy.
derlying data characteristics. Notably, on the ETThl dataset, learnable decomposition occasionally underperforms uni- form masking, particularly at shorter horizons (T = 48). This suggests potential overfitting on smaller datasets.
4.4. Model Efficiency
To demonstrate the computational efficiency of SKOLR, we analyze the model complexity in terms of parameter count, GPU Memory and Running Time. We compare these values with several baseline models the ETTml and weather dataset with sequence length 96 and prediction length 48.
4.3.2. ABLATION: IMPACT FREQUENCY DECOMPOSITION
The improved performance with multiple branches moti- vates further analysis of our learnable frequency decomposi- tion strategy. In Equation 9, the learnable matrix w enables adaptive frequency allocation across branches, in contrast to uniform decomposition (w = 1). Table 5 demonstrates that this learnable approach consistently outperforms uniform allocation across prediction horizons.
Fig. 5 demonstrates SKOLR's computational advantages. On ETTm1, SKOLR achieves the best MSE while us- ing only 3.31 MiB GPU memory. The training speed is also notably faster than other methods. On Weather dataset, SKOLR maintains competitive accuracy, while us- ing significantly less memory and training 4x faster com- pared to the best. This exceptional siitiency-peffrmman trade-off stems from our structured linear operations in Koopman space, avoiding the quadratic complexity of self- attention while maintaining modeling capacity through par- allel branch architecture. The computational efficiency for all datasets can be found Appendix
This adaptive capability is particularly beneficial for datasets with complex temporal patterns (Weather, ECL.), where dif- ferent frequency bands may carry varying importance at different time scales. The learned masks show distinct pat- ters across datasets, suggesting that the model successfully adapts its frequency decomposition strategy based the un-

---

SKOLR: Structured Koopman Operator Linear RNN
Mean Squared Error (MSE)
0.34
DLinear
Mean Squared Error (MSE)
12.05, 18.2MB
0.18
Autorormer 109s, 1339MB
0.32
litarsfarmer 14.95, 3MB
MICN 39.8s, 85. 42.3MB
0.16
DLnnear
MICN 85 95, 03 BMB
65.55, 20.1MB
0.30
PatchTST 32.05, 220MB
Koopa $2.85, 31,91BB
[Trnnfommer
SKOLR
0.14
25. 15. 142.6M8
3.31MB
TimesNet 97.5s, 337
PatchTST 1145, 118/MB
0.28
SKOLR
0.12
13.95, 1.92MB
Keopa 3MB3MB
0.10
20
40
60 80 100
20
40
60
80
100
120
Computation Time per Epoch
Computation Time per Epoch (s)
Figure 5: Model comparison error and training epoch time on P100 GPU. Memory consumption is proportional to circle radius. On ETTml (left) we observe that SKOLR is both the fastest and most accurate method while requiring the smallest memory footprint. On Weather (right) SKOLR is the second best method with much lower training memory and time consumption than the best. For an equivalent budget one could train an ensemble of our approach and obtain better results.
5. Related Work
Although Orvieto et al. (2023) provided insights into the potential connections between the Koopman operator and wide MLP+ linear RNN for representing dynamical sys- tems, this was not the primary focus of their work, and they did not provide equations demonstrating the connection conduct empirical verification. In this work, building on similar insights, we establish an explicit connection by de- riving equations that demonstrate a direct analogy between a structured approximation of a Koopman operator and an architecture consisting of an MLP encoder combined with a linear RNN.
5.1. Koopman Operator-based Time-series Forecasting
Koopman theory (Koopman, 1931) has been applied for modeling and analyzing complex dynamical systems for decades (Mezić, 2005; Brunton et al., 2022). The major advantage of the Koopman operator is that it can represent the dynamical system in the form of linear operator acting on measurement functions (observables). However, learning the operator is challenging because it has infinite dimen- sion. Researchers strive to develop effective strategies for performing finite dimensional approximations; key to this is the selection of good measurement functions. To address this, recent work has explored neural networks for learning the mapping and the approximate operator simultaneously (Li al.. 2017: Lusch al., 2018; Takeishi al., 2017: Yeung al., 2019).
5.2. Deep Learning for Time-Series Forecasting
Time-series forecasting has evolved from statistical mod- els (Makridakis & Hibon, 1997; Hyndman et al., 2008) to deep learning approaches. Previous methods used RNNs (Salinas et al., 2020; Smyl, 2020; Mienye et al., 2024) and CNNs (Bai et al., 2018; Luo et al., 2024) for their ability to capture temporal dependencies. MLP-based archi- tectures (Oreshkin et al., 2020; Challu et al., 2023; Vijay et al., 2023; Wang et al., 2024a) also demonstrated promis- ing performance for forecasting. Recently, transformer ar- chitectures (Nie et al., 2023; Zhang et al., 2024; Hounie et al., 2024; Ilbert al., 2024) introduced powerful atten- tion mechanisms, with innovations in basis functions (Ni et al., 2024) and channel-wist processing (Liu et al., 2024). To address their quadratic complexity, sparse attention vari- ants (Lin et al., 2024) were proposed, but these often strug- gle with capturing long-range dependencies due to infor- mation loss from pruned attention scores. Foundation mod- els (Das et al., 2024; Darlow al., 2024) and unified ap- proaches ( (Woo et al., 2024) have recently emerged. These attempt to mitigate the limitations through pre-training and multi-task learning, but this comes at the cost of dramati- cally increased architectural complexity and computational overhead. To address the complexity challenges in time-
Three recent works address time-series forecasting using Koopman operators. K-Forecast (Lange et al., 2021) uses Koopman theory to handle the nonlinearity in temporal sig- nals and proposes a data-dependent basis for Long-term time- series forecasting. By leveraging predefined measurement functions, KNF (Wang al., 2023b) learns the Koopman operator and attention map to cope with time-series forecast- ing with changing temporal distributions. Koopa (Liu et al., 2023) introduces modular Koopman predictors that sepa- rately address time-variant and time-invariant components via a hicrarchical architecture, using learnable operators for the latter and eDMD (Williams al., 2015) for the for- mer. These prior works rely on hierarchical architectures or complex spectral decompositions to approximate Koopman operators. Our work takes different approach, drawing connection with linear RNNs, paving the way to very efficient and simple forecasting architecture. Our results demonstrate that this strategy leads to improved accuracy with reduced computational overhead and memory.
8

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SKOLR: Structured Koopman Operator Linear RNN
series forecasting, recent state space models (Gu & Dao, 2023) achieve linear complexity, while physics-informed approaches (Verma et al., 2024) enhance interpretability. However, these methods often require complex architec- tures or domain expertise. Our approach offers a balanced solution with principled foundation based on Koopman theory, achieving excellent prediction performance with very low computation and memory requirements.
sequence modeling. arXiv e-prints: arXiv 1803.01271, 2018.
Bevanda, P., Sosnowski, S., and Hirche, Koopman oper- ator dynamical models: Learning, analysis and control. Annu. Rev. Control., 52, January 2021.
Brunton, S. L., Budišié, M., Kaiser, E., and Kutz, N. Modern koopman theory for dynamical systems. SIAM Rev., 64(2), May 2022.
6. Conclusion
Challu, C., Olivares, K. G., Oreshkin, B. N., Ramirez, F. G., Canseco, M. M., and Dubrawski, Nhits: Neural hier- archical interpolation for time series forecasting. In Proc. AAAI Conf. Artif. Intell., 2023.
This work establishes a connection between Koopman op- erator approximation and linear RNNs, showing that time- delayed state representations yield an equivalence between structured Koopman operators and linear RNN updates. Based this, we introduce SKOLR, which integrates learn- able spectral decomposition with a parallelized linear RNN stack giving rise to the structured Koopman operator. By aligning deep learning with Koopman theory, this approach provides a principled and computationally efficient solution for nonlinear time-series modeling. Empirical evaluations on forecasting benchmarks and dynamical systems demon- strate that SKOLR achieves strong predictive performance while maintaining the efficiency of linear RNNs. Future work includes extending this framework to broader dynam- ical systems and exploring alternative spectral representa- tions for enhanced expressivity.
Darlow, L., Deng, Q., Hassan, A., Asenov, M., Singh, R.,
Joosen, A., Barker, A., and Storkey, A. Dam: Towards foundation model for time series forecasting, In Proc. Int. Conf. Learn. Represent. 2024.
Das, A., Kong, W., Sen, R., and Zhou, A decoder-only foundation model for time-series forecasting. In Proc. Int. Conf. Mach. Learn., 2024.
Gu, A. and Dao, T. Mamba: Linear-time sequence modeling with selective state spaces. arXiv preprint arXiv:2312.00752, 2023.
Hounie, I., Porras Valenzuela, J., and Ribeiro, A. Trans- formers with loss shaping constraints for long-term time series forecasting. In Proc. Int. Conf. Mach. Learn., 2024.
Acknowledgement
This research funded by the Natural Sciences and Engi- neering Research Council of Canada (NSERC), [reference number 260250]. Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génic du Canada (CRSNG), [numéro de référence 260250]. Ce projet de recherche #324302 est rendu possible grâce au financement du Fonds de recherche du Quéhec.
Hyndman, R., Koehler, A. B., Ord, J. K., and Snyder, R. D. Forecasting with exponential smoothing: the state space approach. Springer Science & Business Media, 2008
Ilbert, R., Odonnat, A., Feofanov, V., Virmaux, A.. Paolo, G., Palpanas, T., and Redko, Samformer: Unlocking the potential of transformers in time series forecasting with sharpness- aware minimization and channel-wis -wise at- tention. In Proc. Int. Conf. Mach. Learn., 2024.
Impact Statement
Khosravi, M. Representer theorem for learning Koopman operators. IEEE Trans. Autom. Control, 68(5):2995-3010, 2023.
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Kim, T., Kim, J., Tae, Y., Park, C., Choi, J.-H., and Choo, J. Reversible instance normalization for accurate time- series forecasting against distribution shift, Proc. Int. Conf: Learn. Represent., 2022.
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SKOLR: Structured Koopman Operator Linear RNN
A. Experimental Details
A.1. Dataset
We evaluate the performance of our proposed SKOLR on eight widely-used public benchmark datasets, including Weather, Traffic, Electricity, ILI and four ETT datasets (ETTh1, ETTh2, ETTml, ETTm2). Weather is collection of 2020 weather data from 21 meteorological indicators, including air temperature and humidity, provided by the Max-Planck Institute for Biogeochemistry. Traffic is dataset provided by Caltrans Performance Measurement System (PeMS), collecting hourly data of the road occupancy rates measured by different sensors on San Francisco Bay area freeways from California Department of Transportation. Electricity contains hourly time series of the electricity consumption of 321 customers from 2012 to 2014 (Trindade, 2015; Wu et al., 2021). I1.1 dataset contains the weekly time series of ratio of patients seen with ILI and the total number of the patients the United States between 2002 and 2021. ETT datasets are a series of measurements, including load and oil temperature, from electricity transformers between 2016 and 2018, provided by Zhou al. (2021). Following the standard pipelines, the dataset is split into training, validation, and test sets with the ratio of 6:2:2 for four ETT datasets and 7:1:2 for the remaining datasets.
The M4 dataset (Makridakis et al., 2020) consists of 100,000 real-world time series across six frequencies: yearly, quarterly, monthly, weekly, daily, and hourly. It includes data from diverse domains such as finance, economics, demographics, and industry, making it comprehensive benchmark for evaluating forecasting models. Detailed statistics of the datasets are summarized in Table. 6.
Table 6: Forecasting dataset descriptions. The dataset size is organized in (Train, Validation, Test).
Tasks
Dataset
Dim
Prediction Length
Dataset Size
Information (Frequency)
7
{48, 96, 144, 192)
(34465, 11521, 11521)
Electricity (15 mins)
ETTh1, ETTh2
7
(48, 96, 144, 192)
(8545, 2881, 2881)
Electricity (15 mins)
Electricity
321
(48, 96, 144, 192)
(18317, 2633, 5261)
Electricity (Hourly)
Long-term
Traffic
862
[48, 96, 144, 192}
(12185, (757,3509)
Transportation (Hourly)
Weather
21
(48, 96, 144, 192)
(36792, 5271, 10540)
Weather (10 mins)
ILI
7
{24, 36, 48, 60}
(617, 74, 170)
Illness (Weekly)
M4- Ycarly
(23000, O, 23000)
Demographic
M4-Quarterly
1
(24000, O, 24000)
Finance
Short-term
M4-Monthly
18
(48000, O, 48000)
Industry
M4- Weekly
13
(359, 0, 359)
Macro
M4-Daily
(4227, 0, 4227)
Micro
M4- M4-Hourly
48
(414, 0, 414)
Other
A.2. Implementation Details
We implement SKOLR in PyTorch, applying instance-normalizztion and denormalization (Kim et al., 2022) to inputs and predictions respectively. Following the protocol the previous works (Zeng et al., 2023; Nie et al., 2023), SKOLR processes each YI:L.c independently to generate the output ý L1t1:L,,,o and subsequently combine them to form a multivariate forecast. This technique is termed shameel-independence and we omit the variate index m in order simplify the notation in the Section 3.
To improve computational efficiency, we adopt non-overlapping patch tokenization (Nie et al., 2023) before feeding the frequency-decomponn signals I.W. into the linear RNN branches. This reduces the sequence length by factor of P. significantly decreasing computation time while maintaining model effectiveness.
The model architecture employs a single-layer linear RNN to preserve linear state transitions between time steps, with
hhtps://pems.don http:://pmms.oo.c...go.. dot.ca.gov
*Wu et al. (2021) selected 321 of 370 customers from the original dataset in Trindade (2015). This version is widely used in the follow-up works.
12

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SKOLR: Structured Koopman Operator Linear RNN
MLPs using ReLU activation functions in both encoder and decoder components. The patch length adapts the input window length P = T./6. Through grid search, we optimize the branch number N E (2,3,4, 4,8), number of MLP layers M E 21, 3 and dynamic dimension D {128, 256, 512), while maintaining the hidden dimension at H 2D. We train using AdamW W optimizer (Loshchilov, 2017) with learning rate 1 x e and weight decay 5 x em using batch size 32 across all datasets. Complete hyperparameter configurations are detailed in Table 7.
Table 7: Hyperparameters of SKOLR
Datasct
Traffic
EL.C
Weather
ETThl
ETTh2
ETTml
ETTm2
I1.1
Number of Branches N
2
2
2
2
2
2
2
Number of MLP hidden layers M
2
2
2
1
1
Dynamic Dimension D
256
256
128
256
128
256
256
256
Dropout
0.05
0.05
0.2
0.2
0.1
0.2
B. Additional Experimental Results
B.1. Short-term Forecasting
B.1.1. EXPERIMENTAL SETTING
For the short-term forecasting, following the N-BEATS (Oreshkin al., 2020), we adopt the symmetric mean absolute percentage error (sMAPE), mean absolute scaled error (MASE), and overall weighted average (OWA) as the metrics, where OWA is a special metric used in the M4 competition. These metrics can be calculated as follows:
200
Y
Y
sMAPE
T
Y,
+ |
i=1
|Y, Yil
MASE T
Σ
Ej=m+1 IY, Yz-ml
1= T-m
sMAPE
MASE
OWA
2
SMAPE
MASENaive
where m is the periodicity of the data. Y, y E TxC are the ground truth and prediction results of the future with T time points and C dimensions. Y; represents the i-th future time point.
We compare SKOLR against state-of-ther the art models the M4 competition dataset, which contains six diverse domains (Yearly, Quarterly, Monthly, Weekly, Daily, Hourly) with prediction horizons ranging from 6 to 48 steps, as shown in Table 6. The baselines includes: N-BEATS (Oreshkin et al., 2020), N-HiTS (Challu et al., 2023), PatchTST (Nie et al., 2023), TimesNet (Wu et al., 2023), DLinear (Zeng et al., 2023), MICN (Wang et al., 2023a), KNF (Wang et al., 2023b), FiLM (Zhou et al., 2022), Autoformer (Wu et al., 2021), and KooPA (Liu et al., 2023). These models are not specifically designed for long-term forecasting, but also generalize well on short-term tasks.
1.2. RESULTS
Table 8 demonstrates SKOLR's effectiveness in handling diverse temporal patterns across M4 domains. At yearly, quarterly and monthly predictions, where data exhibits strong seasonality and multiple frequency components, SKOLR's parallel branch design allows simultancous tracking of different temporal scales. Our structured Koopman operator design proves particularly powerful for the M4 dataset, which contains richer dynamic information than typical long-term forecasting benchmarks, resulting in consistently outperforming both traditional forecasting models A-BEETS, N-HiTS) and other Koopman- based approaches (KooPA) across all evaluation metrics.
13

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SKOLR: Structured Koopman Operator Linear RNN
Table 8: Comparison of Models for short-term prediction. Best results and second best results are highlighted in red and blue respectively.
M4
Metric
SKOLR
KooPA N-HiTS N-BEATS PatchTST TimesNet DLinear
MICN
KNF
FiLM
Autoformer
sMAPE
13.291
13.352
13.371
13.466
13.517
13.394
13.866
14.532
13.986
14.012
14.786
Year
MASE
2.996
2.997
3.025
3.059
3.031
3.004
3.006
3.359
3.029
3.071
3,349
OWA
0.784
0.786
0.790
0.797
0.795
0.787
0.802
0.867
0.804
0.815
0.874
MMAPE
9.986
10.159
10.454
10.074
10,847
10.101
10.689
11.395
10,343
10.758
12.125
Quarter
MASE
1.166
1.189
1.219
1.163
1.315
1.183
1.294
1,379
1.202
1.306
1.483
OWA
0.878
0.895
0.919
0.881
0.972
0.890
0.957
1.020
0.965
0.905
1.091
SMAPE
12.536
12.730
12.794
12.801
14.584
12.866
13.372
13.829
12.894
13.377
15,530
Month
MASE
0.921
0.953
0,960
0.955
1.169
0.964
1.014
1.082
1.023
1.021
1.277
OWA
0.867
0.901
0.895
0.893
1.055
0.894
0.940
0.988
0.985
0.944
1.139
MAPE
4.652
4,861
4.696
5,008
6,184
4.982
4.894
6.151
4.753
5.259
5.841
Others
MASE
3.233
3.124
3.130
3,443
4,818
3.323
3,358
4.263
3.138
3.608
4.308
OWA
0.999
1.004
0.988
1.070
1.140
1.048
1.044
1,319
1.019
1.122
1.294
sMAPE
11.704
11.863
11.960
11.910
13.022
11,930
12.418
13.023
12.126
12.489
14.057
Average MASE
1.572
1.595
1.606
1.613
1,814
1.597
1.656
1,836
1.641
1.690
1,954
OWA
0.843
0.858
0.861
0.862
0.954
0.867
0.891
0.960
0.874
0.902
1.029
Table 9: Prediction results benchmark datasets with L = 96. Best results and second best results are highlighted in red and blue respectively.
Dataset
Koopa MSE
l'Iansformer MSE
PaichTST
Crossfommr
TIDE
TimesNet
Dlinear MSE
SCINer
Stationary MSE
Autaformer
MSE
MAE
MAE
MAB
MSE
MAB
MSE
MAE
MSE
MAE
MSE
MAE
MAE
MSE
MAE
MAE
MSE
MAB
0.313
0.356
0.330
0.368
0.334
0.368
0.309
0.367
0.404
0.426
0.364
0.387
0.338
0.375
0.345
0.372
0.418
0.438
0.386
0.398
0.505
0.175
192
0.359
0.384
0.385
0.395
0.377
0.391
0.367
0.383
0.450
0.451
0.398
0.404
0.374
0.387
0.380
0.389
0.439
0.450
0.459
0.444
0.553
0.496
BTTml
356
0.389
0.406
0.402
0.413
0.426
0.420
0.500
0410
0,532
0.515
0.428
0.425
0.410
0.411
0.413
0.433
0.490
0.485
0.495
0.464
0.621
0.537
0.643
0.472
0.419
0.459
0.439
0.589
0:461
0.478
0.450
0.474
0.595
0.585
0.671
0,175
0.256
0.181
0.263
0.130
0.264
0.259
0.287
0.366
0:207
0.305
0.187
0.267
0.193
0.292
0.286
0.377
0.192
0.255
0.339
0.239
0.300
0,248
0.308
0.309
0.241
0.302
0,414
0.692
0.290
0.364
0.249
0.309
0.284
0,362
0.399
0.445
0.280
0.281
0.240
336
0.302
0.341
0.303
0.345
0.311
0.348
0.305
01543
0.597
0.542
0.377
0.422
0.521
0.351
0.369
0.427
0.657
0.591
0.334
0.361
0.339
0.372
720
0.398
0.396
0.403
0.401
0.412
0.407
0.502
0.400
L.730
1.042
0.558
0.524
0.408
0.403
0.554
0.522
0.960
0.735
0.417
0.413
0.433
0.432
96
0.371
0.397
0.401
0.413
0.386
0.405
0.414
0.419
0.423
0.448
0.479
0.464
0.384
0.402
0.386
0,400
0.654
0.599
0.513
0.491
01440
0.49
BTTH1
192
0.423
0.427
0.453
0.419
0.461
0.439
0.441
0.458
0.436
0.660
0.445
0.466
0:471
0.474
0.546
0.525
0.515
0.492
0.416
0.469
0.499
0.481
0.437
0.459
0.43
0.719
0.778
0.631
0.534
0.588
0.504
0,500
0.422
720
0.099
0.484
0.484
0.472
0.503
0.191
0.500
0.458
0.653
0.621
0.594
0.558
0.521
0.500
0.519
0.516
0.836
0.699
0.643
0.616
0.514
0.512
0.293
0.344
0.316
0.361
0.297
0.349
0.902
0.348
0.745
0.584
0.400
0.440
0.340
0.374
0.333
0,387
0.207
0.621
0.476
0.346
BTTb2
192
0.370
0.384
0.384
0.405
0.380
0.400
0,383
0.400
0,877
0.656
0.528
0.509
0,402
0.414
0.477
0.476
0.860
0.689
0.512
0.493
0.456
0.522
336
0.410
0.428
0.423
0.438
0.488
0.422
0.436
0.43
1.043
0.731
0.643
0:571
0.452
0.452
0.594
0.541
1.000
0.744
0.552
0.551
01432
0.466
0.431
0.446
0.458
0.445
0.446
1.104
0.763
0.679
0.462
0.468
1.249
0.562
0.515
0.153
0.346
0.146
0.344
0.148
0.340
0.181
0.270
0.219
0.314
0.237
0.329
0.168
0.272
0.197
0.262
0.347
0.345
0.169
0.373
0.201
0.317
BCL
336
0.169
0.168
0.259
0.28
0.169
0189
0.266
0.285
0.178
0.253
0.269
0.188
0.00
0.274
0.291
0.231
0,266
0.337
0.322
0.236
0.249
0.330
0.344
0.198
0.184
0.300
0.196
0.209
0.285
0,301
0.257
0.269
0.369
0.182
0.200
0.304
0.231
0.222
0.334
0.338
720
0,250
0.318
0.226
0.314
0.2
0.225
0.117
0.246
0.304
0.280
0.363
0.284
0.375
0.229
0.330
0.345
0.333
0.299
0,390
0.332
0.321
0.254
0.361
96
0.427
0.270
0.462
0.290
0.395
0.268
0.462
0.295
0.522
0,290
0.805
0.493
0.593
0.321
0.650
0.396
0.788
0.499
0612
0.338
0.613
0.388
Traffic
192
12455
1.289
0.566
0.386
0.417
0.276
0,406
0.296
0,530
0.293
0.756
0.474
0.617
0.336
0.598
0.570
0.789
0,501
0.613
0.340
0.116
0.382
0.472
0.24%
0.351
0.433
0.283
0.482
0.304
0.558
0.305
0.762
0.477
0.629
0.336
0.605
0.797
0,508
0.618
0.622
0.519
0.336
0.552
0.346
0.667
0.302
0.514
0.322
0.589
0.328
0.719
0.449
0.640
0.350
0.645
0.394
0.841
0.525
0.633
0.600
0.408
0.207
0.202
0.214
0.177
0.218
0.158
0.230
0.202
0.261
0.172
0.220
0.196
0.255
0.221
0.173
0.223
0.266
0.336
Weather
336
0.266
0.208
0.249
0.292
0,266
0.209
0.251
0.290
0.278
0.221
0.296
0.354
0,278
0.225
0.297
0.250
0,272
0.206
0.335
0:277
0.287
0.242
0.335
0.298
0.250
0.319
0.261
0.306
0,283
0.257
0.206
0.335
0.261
0.309
0.340
0,378
0.321
0.345
0.338
0.285
0.359
0.307
0.367
0.395
720
0.344
0,343
0.350
0.348
0.358
0.347
0.354
0.348
0.398
0.418
0.351
0.386
0.365
0.359
0.345
0.381
0.377
0.427
0.414
0.110
0.419
0.428
B.2. Results for a shorter look-back window
For a fair comparison, we also conduct the experiments under the L = 96 setting that is the default for iTrrnsformmer TimesNet, and other transformer-basee models, except PatchTST. For the shorter look-back window, we use the patch length
P = 16 to obtain the tokens for SKOLR. By default, SKOLR follow the hyperparameters in Table 7. As shown in Table 9, SKOLR emerges as the leading performer, achieving Rank 1 or 2 in 24 cases out of 28 cases in terms of MSE and 25 cases in terms of MAE
14

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SKOLR: Structured Koopman Operator Linear RNN
Table 10: Model performance across different datasets with mean + standard deviation for MSE and MAE metrics.
Dataset
Models
MSE
MAE
0.137 0.0003
0.229 0.0003
96
0.132 + 0.0005
0.225 + 0.0004
ECL.
144
0.143 + 0.0001
0.236 +: 0.0001
192
0.149 + 0.0001
0.244 + 0.0001
48
0.400 + 0.0003
0.258 +: 0.0040
96
0.368 -. 0.0007
0.248 1: 0.0007
Traffic
144
0.375 0.0003
0.255 0.0002
192
0.377 + 0.0003
0.256 - 0.0002
0.131 0.0009
0.170 0.0008
96
0,154 + 0.0015
0,202 + 0.0015
Weather
144
0.172 + 0.0009
0,220 + 0.0006
192
0.193 + 0.0004
0,241 = 0.0005
48
0.280 + 0.0013
0.330 + 0.0015
96
0.287 + 0.0003
0.340 + 0.0001
ETTml
144
0.313 0.0020
0.361 0.0023
192
0.328 + 0.0019
0.373 - 0.0018
0.134 0.0011
0.228 0.0007
96
0.171 +1 0.0015
0.255 1 0.0013
ETTm2
144
0.209 + 0.0014
0.283 + 0.0014
192
0.241 + 0.0013
0.304 - 0.0015
48
0.333 + 0.0009
0.373 + 0.0007
96
0.371 + 0.0011
0.398 + 0.0008
ETTh1
144
0.405 0.0019
0.417 = 0.0020
192
0.422 + 0.0030
0.432 + 0.0034
0.238 0.0012
0.306 0.0004
96
0.299 + 0.0034
0.352 + 0.0042
ETTh2
144
0.335 + 0.0042
0.377 + 0.0048
192
0.365 + 0.0033
0.397 + 0.0040
24
1.556 1 0.0213
0.760 1: 0.0159
36
1.462 + 0.0711
0.728 + 0.0676
ILI
1.537 0.0038
0.798 0.0030
60
2.187 + 0.0435
0.995 - 0.0498
B.3. Experimental Variability
Table 10 reports standard deviation (std) across independent runs for all datasets and forecast horizons. The low stds (<0.003 for most datasets) demonstrate the consistency of SKOLR's performance.
B.4. Comparison with Orvieto et al. (2023)
The Linear Recurrent Unit (LRU) presented by Orvieto et al. (2023) derived from vanilla recurrent neural networks (RNNs) through a sequence of principled modifications including linearization of the recurrence, diagonalization, exponential parametrization for stability, and forward- pass normalization. These changes yield a model that can match the performance of recent deep state-space models (SSMs) such as S4 and S5 on benchmarks like the Long Range Arena (Tay et al., 2021), without relying discretization of continuous-tine dynamics.
We implement our code in Porch. Our implementation follows from the JAX pseudocode presented in the original paper's appendix (Orvieto et al., 2023). Additionally, we consulted a community implementation in PyTorch The LRU is trained using the Adam optimizer with no weight decay applied to the recurrent parameters. Learning rates are selected via grid search on a logarithmic scale. All experiments use networks of 6 LRU layers with residual and normalization layers between blocks and final linear output layer and 64-dimensional hidden state.
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15

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SKOLR: Structured Koopman Operator Linear RNN
Table 11: Performance comparison of LRU, Koopa and SKOLR on non-linear dynamical systems (NLDS)
SKOLR
KooPA
LRU
Dataset
MSE
MAE
MSE
MAE
MSE
MAE
Pendulum
0.0001
0.0083
0.0039
0.0470
0.0572
0.0242
Duffing
0.0047
0.0518
0.0365
0.1479
0.0573
0.5970
Lotka-Vole Volterra
0.0018
0.0354
0.0178
0.1050
0.2058
0.3779
Lorenz '3
0.9740
0.7941
1.0937
0.8325
1.1905
0.8932
Whereas focus in SKOLR is time-series forecasting, Orvieto al. (2023) target long-range reasoning. Although it is possible to convert their architecture to address forecasting, performance suffers because it is not the design goal, as shown in Table 11.
C. Model Efficiency
C.1. Theoretical Complexity Analysis
SKOLR achieves computational efficiency through its structured design and linear operations. For a time series of length with patch length P, embedding dimension D, and N branches, analyze both time and space complexity.
The time complexity of SKOLR consists of several components: spectral decomposition, encoder/decoder MLPs, and linear RNN computation. If we perform a single FFT operation OLL log L) followed by branch-specific frequency filtering, the main computational cost comes from encoder/decoder MLPs O(N x (L/P) x D²) and linear RNN computation O(N x (L/P) x D²), giving a total time complexity of O(N x (L/P) x D²). The memory complexity includes model parameters O(N x D2) and activation memory O(N (L/P) x D).
Our structured approach with N branches provides substantial efficiency gains compared to non-structured approach with equivalent representational capacity. For non-structured model with dimension D' = N D, the time complexity would be ) and memory complexity D?) This represents an N-fold increase in computational requirements. For example, with N = 16 branches, our structured approach requires approximately 16x fewer parameters and operations while maintaining equivalent or better modeling capacity, as shown in Section 4.3.1. Compared to transformer based approaches with time complexity O(LL/)²² D + (L/P) x D²) and memory complexity ((LLP)) + (L/P) x D), SKOLR demonstrates a fundamental advantage for long sequences by avoiding the quadratic scaling with sequence length.
O(LL/P) x D²)
C.2. Parallel Computing
SKOLR further benefits from parallel processing capabilities. The N separate branches can be processed completely independently, reducing the effective time complexity O(LLP)) D²) with sufficient parallel resources. This branch- level parallelism implemented in current code.
In future work, we plan to implement additional parallelization of the linear RNN computation itself. Since our RNN has no activation functions, we can express the hidden state evolution for each branch with sequence length L/ P in closed form: hu = ((xx) L/P W "g({)-)) where W indicates s applications of W (the state transition matrix). This formulation allows to compute all hidden states simultaneously through efficient matrix operations, potentially reducing the time complexity further O log(L/P) + (L/P)² x D) per branch.
For time series with patching where L/P P < D, this approach achieves significant speedups by eliminating the sequential dependency in RNN computation. With both branch and RNN parallelism implemented, SKOLR can achieve greater computational efficiency while maintaining its forecasting performance.
C.3. Computational Efficiency
We provided efficiency results on the ETTm2 and Traffic datasets in Fig. 5. We have expanded our evaluation across additional datasets to offer a more comprehensive assessment in Table 12. In all datasets, the proposed architecture provides a compelling trade-off between efficiency and accuracy compared to baselines.
16

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SKOLR: Structured Koopman Operator Linear RNN
Table 12: Model Efficiency and Performance Comparison for Different Datasets with T = 96. Parameters (Params) are
measured in millions (M), GPU memory (GPU) in MiB, computation time per epoch in seconds (s) on NVIDIA V100 GPU with batch size 32.
(a) Traffic
(b) Electricity
Model
Params (M) GPU(MiB)
Time (8)
MSE
Model
Params (M)
GPU(MiB)
Time (s)
MSE
Autoformer
14.914
18.811
51.0
0.668
Autoformer
11.214
17.373
68.7
0.182
iTransformer
6.405
62.710
126.0
0.388
iTransformer
4,957
86.478
58.6
0.134
PatchTST
3.755
22.132
1042.0
0,413
PatchTST
6,904
73.517
1231.0
0.143
MICN
236.151
32.310
84.0
0,511
MICN
6,635
32.668
18.0
0.165
TimesNet
30.170
111.998
6563.0
0.611
TimesNet
15.037
33.435
11351.0
0.170
DLinear
0.009
12.861
7.7
0.485
DLinear
0.019
76.016
6.8
0.153
Koopa
5.429
50.335
25.5
0.401
Koopa
4.076
31.067
33.1
0.136
SKOLR
1.479
5.915
216.0
0.368
SKOLR
1.541
6.163
99.1
0.132
(c) ETThl
(d) ETTm2
Model
Params (M)
GPU(MiB)
Time (s)
MSE
Model
Params (M)
GPU(MiB)
Time (s)
MSE
Autoformer
10.536
16,523
29.5
0.634
Autoformer
10.536
14.599
152.6
0.241
iTransformer
0.237
27.245
4.1
0.393
iTransformer
0.237
27.245
13.1
0.177
PatchTST
3.752
22.018
8.5
0.372
PatchTST
10.056
39.910
980.0
0.171
MICN
252.001
65.974
21.1
0.406
MICN
252.001
65.974
84.2
0.197
TimesNet
0.605
26.053
22.1
0.411
TimesNet
1.192
34.783
113.0
0.187
DLinear
0.140
26,440
0.6
0.379
DLinear
18.291
9.312
1.9
0.172
Koopa
0.135
31.951
10.1
0.371
Koopa
0.135
31.951
48.2
0.171
SKOLR
0.429
1.717
2.8
0.371
SKOLR
0.429
1.717
12.6
0.171
D. Additional Analysis
D.1. Scaling Up Forecast Horizon
We have conducted experiments to explore performance in the setting where forecast horizon increased at test-time. In this experiment, SKOLR and Koopa were evaluated by scaling up from the training horizon (Tr) a larger test horizon (Tw). Unlike Koopa (Liu al., 2023), SKOLR does incorporate operator adaptation (OA) mechanism to update its Koopman operator using incoming ground truth. Instead, our architecture possesses natural recursive structure that enables straightforward extension to longer horizons. Even when weights are trained to minimize a loss function specified over given horizon, the algorithm can be recursively applied to predict over extended periods.
As demonstrated in Table 13, SKOLR maintains competitive performance without requiring additional adaptation mecha- nisms. The structured Koopman operator and linear RNN design enable robust long-term predictions, with error percentages remaining comparable to Koopa OA across various datasets. This demonstrates SKOLR's inherent capability handle extended forecast horizons efficiently through its recursive architecture.
D.2. Ablation Study
We have also conducted more comprehensive ablation study on the design elements of SKOLR. As shown in Table 14, we compare full SKOLR model with two ablated variants: (1) "w/o Structure": no structured decomposition, using single branch with dimension (NxD); (2) "w/o Spectral Encoder": no learnable frequency decomposition, while maintaining the multi-branch structure.
The results show that both components contribute meaningfully. Removing the structured decomposition leads to perfor- mance degradation on 27/32 tasks, with notable declines on ETTh1 and ILI, while increasing computational overhead. Similarly, removing the spectral encoder impacts performance on 23/32 tasks, though with smaller overall effect.
17

---

SKOLR: Structured Koopman Operator Linear RNN
Table 13: Scaling up forecast horizon: (T.rr, T_te)= (24, 48) for ILI and (T.tr, T_te) (48, 144) for others. Koopa and
SKOLR conducts vanilla rolling forecast and Koopa OA has operator adaptation.
ETTh2
ECL
Traffic
Weather
(ADF
1.135
(ADF
-5,406)
(ADF -8.483)
(ADF -15.-15.06))
(ADF -26.666.661
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MSE
MAE
Koopa (T_tr)
0.226
0.300
1.621
0.800
0.130
0.234
0.415
0.274
0.126
0.168
Koopa
0.437
0.429
2.836
1.065
0.199
0.298
0.709
0.437
0.237
0.276
Error(+ %)
93%
43%
75%
33%
53%
27%
71%
59%
88%
64%
Koopa OA
0.372
0.404
2.427
0.907
0.182
0.271
0.699
0.426
0.225
0.264
Error(+ %)
65%
35%
50%
13%
40%
16%
68%
55%
79%
57%
SKOLR (T.tr)
0.238
0.306
1,556
0.760
0.137
0.229
0.400
0.258
0.131
0.170
SKOLR (T te)
0.393
0.402
2.392
0.958
0.204
0.289
0.612
0.383
0.222
0.257
Error(+ %)
65%
31%
54%
26%
49%
26%
53%
48%
69%
51%
Table 14: Ablation study comparing SKOLR with versions without structure and without spectral encoder
SKOLR
w/o Structure
w/o Spectral Encoder MSE
Dataset
MSE
MAE
MSE
MAE
MAE
48
0.137
0.229
0.148
0.238
0.149
0.238
96
0.132
0.225
0.135
0.228
0.133
0.227
ECL
144
0.143
0.236
0.146
0.241
0.142
0,235
192
0.149
0.244
0.150
0.245
0.148
0.243
48
0,400
0.258
0.395
0.255
0.397
0.257
96
0.368
0.248
0.367
0.249
0.369
0.249
Traffic
144
0.375
0.255
0.375
0.255
0.375
0.255
192
0.377
0,256
0.378
0.256
0.377
0.256
48
0.131
0.170
0.134
0.173
0.134
0.172
96
0.154
0,202
0.157
0.203
0.158
0.202
Weather
144
0.172
0.220
0.177
0.225
0.175
0.221
192
0.193
0.241
0.195
0.242
0.197
0.244
48
0.280
0.330
0,284
0.334
0.282
0.332
96
0.287
0.340
0.292
0.343
0.291
0.342
ETTml
144
0.313
0.361
0.325
0.365
0.319
0.361
192
0.328
0.373
0.332
0.372
0.332
0.372
0.134
0.228
0.135
0,229
0.162
0,259
96
0.171
0.255
0.174
0.259
0.169
0,253
ETTm2
144
0.209
0,283
0.206
0.280
0.209
0.282
192
0.241
0.304
0.241
0.305
0.230
0.299
48
0.333
0.373
0.338
0.377
0.336
0.374
96
0.371
0.398
0.387
0,408
0.373
0.399
ETTh1
144
0.405
0.417
0.414
0,423
0.410
0.420
192
0.422
0,432
0.409
0.421
0.413
0.422
0.238
0.306
0.233
0.304
0.239
0,305
96
0.299
0.352
0.301
0.350
0.303
0.350
ETTh2
144
0.335
0.377
0.341
0.382
0.337
0.381
192
0.365
0.397
0.370
0.398
0.370
0.401
24
1.556
0.760
1.795
0.842
1.522
0.741
36
1.462
0.728
1,990
0.889
1.496
0.734
ILI
48
1.537
0.798
1.875
0.909
1,571
0.810
2.187
0.995
2.407
1.056
2.263
0,999
18

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SKOLR: Structured Koopman Operator Linear RNN
FFT
Signal
Branch Input
0
50
25
Input
True
Pred
75
50
0
25
-1
Branch 2
75
0
-1
50
0
100
200
300 Steps
400
500
600
Frequency (μHZ)
Figure 6: Analysis of SKOLR's branch-wise behavior on ETTm2 feature 6: (a) frequency decomposition and (b) prediction performance.
FFT
Signal
Branch 2 Branch Input
200
2.5
M NMM
100
0.0
Input
200
2.5
100
0.0
200
2.5
W AM
100
0.0
100
200
300 Steps
400
500
600
20
40
60
Frequency (uHz)
Figure 7: Analysis of SKOLR's branch-wise behavior on Electricity feature 11: (a) frequency decomposition and (b) prediction performance.
D.3. Branch-wise Visualization
We add examples from ETTm2 and Electricity in order to analyze SKOLR's branch-wise behavior. The figures show distinct frequency specializations. In Fig. 6, SKOLR decomposes the time series into complementary frequency components, with Branch focusing on lower, distributed frequencies, and Branch 2 capturing more specific dominant frequency peaks. In Fig. 7, Branch shows higher amplitudes across most of the very low (0 20 frequency components compared to Branch 2. The time-domain plots demonstrate how these spectral differences translate into signal reconstruction; Branch focuses more on prediction of the higher-frequency components of the time series.
D.4. Analysis Error Accumulation
Time-series forecasting has two main prediction methods: direct prediction, which forecasts the entire horizon at once but and cannot extend the prediction horizon after training, and recursive prediction, which iteratively uses predictions as inputs but may suffer from eиrо accumulation over long sequences.
In SKOLR, of operating at the individual timestep level, we work with patches of multiple timesteps, directly predicting all values within each patch while only applying recursion between patches. This dramatically reduces the number of recursive steps (e.g., from 720 to just 5 with patch length 144), controlling error accumulation. Additionally, this method also reduces complexity to O)))) from RNN standard timestamp- based approaches (O(L) while maintaining the core principle of Koopman operator theory.
patching approach (Appendix A.2) to create effective middle ground between these methods. Instead
19

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SKOLR: Structured Koopman Operator Linear RNN
Step-wise MSE Evolution
SKOLR P=16
0.5
SKOLR P-24
SKOLR P-48
Mean Squared Error
SKOLR P=144
0.4
SKOLR P=240
[Tranffomer
0.3
0.2
200
500
Time Steps in Prediction Horizon
Figure 8: Error progression across 720 time steps: SKOLR (multiple patch sizes) vs. iTrrnsformer
To better address this question, add experiment longer horizon L 720, T 720 on dataset ETTm2 with varying iteration steps. We vary the patch length P {16, 24, 48, 144, 240) of the SKOLR model to see the difference performance caused by the number of iterations. SKOLR with / = 16 requires 45 recursive steps, while / = 240 needs only 3, yet they maintain comparable enror profiles in Fig. 8. This empirically demonstrates that SKOLR's s patch-based based approach effectively controls error accumulation, even with increased recursion.
We also compare SKOLR with iTransformer (Liu al., 2024), which performs direct prediction without recurrence. Both models show similar patterns of error increase with longer horizons, suggesting that this modest increase is inherent to all forecasting approaches when extending the prediction range, rather than being caused by recurrent error accumulation.
D.5. Analysis Koopman operator eigenvalue
The Koopman modes derived through eigendecompossiionn of the RNN weight matrices M,. These modes represent dynamical patterns in the data. Each mode captures specific components of the time series. The stability and oscillatory behavior of each mode is determined by the corresponding eigenvalue's position in the complex plane. The eigenvalue plots in Fig. 9 show that each branch learns complementary spectral properties, with all eigenvalues within the unit circle, indicating stable dynamics. Branch shows concentration at magnitude 0.4, while Branch exhibits a more uniform distribution.
Moreover, this observation provides some reassurance against error accumulation concerns, as divergence is more likely for unstable systems. Our learned system's stability encourages error effects to naturally decay over time during forward prediction steps rather than compounding.
E. State Prediction for Nonlinear Dynamical Systems (NLDS)
generated datasets for four nonlinear dynamical systems (NLDS) to evaluate the performance of Koopman- -based models in state prediction tasks. Each dataset contains trajectory with a total of 20000 time steps. The first 14000 steps were designated for training, 2000 steps used for validation, while the remaining 4,000 steps used for inference. Below, we describe the generation process for each system:
Pendulum: A simple nonlinear pendulum system described by its angular displacement and velocity. The trajectory was initialized with random a angle and angular velocity, with updates computed using the equations of motion under gravity. The system being simulated a simple pendulum, consisting massless rod of length (. The pendulum swings in a two-dimensional plane under the influence of gravity, with the gravitational acceleration g. The motion of the pendulum governed by the equation of motion:
mass m attached to the end of a rigid,
20

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SKOLR: Structured Koopman Operator Linear RNN
Koopman Operator Eigenvalues: traffic
Branch 1 Eigenvalues
Branch 2 Eigenvalues
Combined Eigenvalues
1.0
Branch
Branch
Branch 1
Imaginary Part
Imaginary Part
Imaginary Part
Branch
0.5
0.5
0.5
0.0
0.0
0.0
-0.5
-0.5
-0.5
-1.0
-1.0
-1
-1
0
Real Part
Real Part
Real Part
(a)
Distribution of Eigenvalue Magnitudes: traffic
Branch 1
Branch 2
Unit Magnitude
20
Frequency
15
10
0.2
0.4
0.6
Magnitude (IX)
(b)
Figure 9: Koopman operator eigenvalue analysis for SKOLR on the Traffic dataset
8 8"(t) + sin()(t)) = 0
(17)
where:
(tt) is the angular displacement of the pendulum at time 6,
(() is the angular acceleration,
9 is the acceleration due gravity (9.81 m/s²),
7 is the length of the pendulum (set to 1.0
The system is further characterized by its initial conditions:
The initial angle dg. which randomly chosen from a uniform distribution between -x and x.
The initial angular velocity wh. which is randomly chosen from uniform distribution between 1 rad/s and 1 rad/s.
The motion of the pendulum is modeled using numerical methods, specifically the Euler method, which approximates the solution of the system of equations over discrete time steps.
Duffing Oscillator: A nonlinear oscillator characterized by damping and cubic stiffness terms. Trajectories were generated using randomized initial positions and velocities, with dynamics influenced by an external periodic driving force. The system modeled by the code is Duffing oscillator, a type of nonlinear second-order differential equation
21

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SKOLR: Structured Koopman Operator Linear RNN
commonly used to describe systems with nonlinear restoring forces and damping. The equation of motion for the Duffing oscillator is given by:
- 5á + ax Bxx² 7 cos(wt)
where æ(() represents the displacement of the oscillator, v(() =i(() represents its velocity, and / is time. The parameters of the system are: = 1.0 (linear stiffness), B = 5.0 (nonlinear stiffness), 5 = 0.3 (damping coefficient), 8.0 (driving force amplitude), and w = 0.5 (angular frequency of the driving force). The system undergoes periodic driving forces, and its motion influenced by both the nonlinear restoring force and damping. The motion of the oscillator is simulated by numerically integrating the equations motion using simple time-stepping method, where the time step, and the initial conditions for a and y are randomly selected within a small range. The system's 's behavior is characterized by chaotic dynamics for the chosen parameter values.
Lotka- Volterra: predator-pre prey population model, where the prey and predator populations interact dynamically, Trajectories were initialized with random population sizes, and updates followed the Lotka- Volterra equations. The equations governing the Lotka- Volterra predator-pre prey model are given by:
Lotka-Voltera
d.N prey
OxNprey prey
3 3N N predator
dt
d. predator
8N prey Npredator 7 predater
dt
where:
Npryy is the population of the prey species,
N Predttto is the population of the predator species,
is the natural growth rate of the prey.
B is the predation rate (rate at which predators kill prey),
8 is the rate at which predators increase due consuming prey,
- is the natural death rate of the predator.
In this model, the prey species grows exponentially in the absence of predators, and the predator species declines expo- nentially in the absence of prey. The interaction between the species causes cyclical fluctuations their populations.
We implement this model by numerically integrating the differential equations using a simple Euler method. The process starts by initializing the prey and predator populations randomly within given range. The parameters 1.1, 3 0.4, 8 0.1, and 7 0.4 are then used to update the populations each time step.
Lorenz '63: A chaotic system described by three variables: æ, y, and 2. Each trajectory is started with randomized initial conditions, and updated using the Lorenz equations with standard parameters. The equations governing the Lorenz system are given by:
dr
aly x)
dt
dy
= x(p 2)
dt
dz
xy B2
dt
where:
y. and represent the state variables of the system, typically interpreted as the variables describing the convection rolls in the atmosphere, is the Prandtl number, a measure of the fluid's viscosity, set 10.0,
22

---

SKOLR: Structured Koopman Operator Linear RNN
is the Rayleigh number, representing the temperature difference between the top and bottom of the fluid, 28.0,
- 3 is a geometric factor, set to 3
The Lorenz system exhibits chaotic behavior for these parameter values, meaning that small differences in initial conditions lead vastly different outcomes over time. In the simulation, the system of differential equations is solved using the Fuler method over series of time steps. A visualization of the system is shown in Fig. 10.
Lorenz System Trajectory
Lorenz System Trajectory
50
40
30 2
20
10
20.15_10_5
y
10
15
20
20
Figure 10: Lorentz '63 system plotted in 3D
23


Reviews:
[{"review_id": "AzHsphWKc7", "reviewer": "Reviewer_kXAS", "summary": "This paper proposes a new linear RNN for time-series forecasting inspired by Koopman operator theory. The problem setup is, for an autonomous dynamical system $\\mathbf{x}\\_{k+1}=F(\\mathbf{x}\\_k)$ with observable $\\mathbf{y}\\_k=h(\\mathbf{x}\\_k)$ for an unknown $h(\\cdot)$, to condition on a partial trajectory $\\mathbf{y}\\_1,...,\\mathbf{y}\\_L$ of a fixed length $L$ and forecast $T$ steps of the future $\\mathbf{y}\\_{L+1},...,\\mathbf{y}\\_{L+T}$. The authors start with the standard form of a linear RNN which evolves its hidden state $\\mathbf{h}_k$ conditioned on an input sequence $\\mathbf{v}_k$, and consider the case where $\\mathbf{v}_k$ is pointwise mapped from $\\mathbf{y}_k$. This reduces the linear RNN to $\\mathbf{h}\\_k=\\sum\\_{s=1}^k\\mathbf{W}^{k-s} g(\\mathbf{y}\\_s)$ for some $g(\\cdot)$. Then, the authors propose to construct an approximate state as a stack of past $L$ measurements  $\\tilde{\\mathbf{x}}\\_k \\coloneqq [\\mathbf{y}\\_{k-L+1},...,\\mathbf{y}\\_k]$ and learn the dynamics $\\tilde{\\mathbf{x}}\\_k=\\tilde{F}(\\tilde{\\mathbf{x}}\\_{k-1})$ by jointly learning $g(\\cdot)$ on $\\tilde{\\mathbf{x}}\\_k$ and a matrix $\\mathbf{M}$ which parameterizes the dynamics of $g(\\tilde{\\mathbf{x}}\\_k)$, using a similar objective to the standard EDMD, but with a structured constraint on $\\mathbf{M}$ such that the learned dynamics can be always written as $g(\\tilde{\\mathbf{x}}\\_k) = \\mathbf{h}\\_k=\\sum\\_{s=1}^k\\mathbf{W}^{k-s} g(\\mathbf{y}\\_s)$. Basically, this constraint means that $\\mathbf{M}$ is a blockwise diagonal matrix with each block given as a power of some learnable matrix $\\mathbf{W}$. The authors parameterize $g(\\cdot)$ as trainable gating in frequency domain (FFT -> trainable gate per frequency -> IFFT) followed by pointwise MLP, at multiple \"heads\". Then at each head the structured matrix $\\mathbf{M}$ is separately parameterized. The decoder is simply a positionwise MLP. The authors report state-of-the-art performance of the proposed architecture on 8 benchmark datasets, using $L=2T$, and 4 physical dynamical systems. Further analysis and ablation study on the role of each component and hyperparameter are provided.\n\n## update after rebuttal\n\nThe following is my revised understanding of the paper:\n\n- The paper proposes a new linear RNN for time-series forecasting. The key components proposed are block-diagonal state-transition matrix and learnable frequency-domain gating. The former is inspired by Koopman operator theory for deterministic nonlinear dynamical systems and the EDMD algorithm. The latter is inspired by existing works in time-series modeling leveraging frequency-domain analysis components.\n\n- Empirically, the proposed method overall shows a competitive performance compared to the state-of-the-art Koopa, and at the same time has a good time and space computational efficiency.\n\n- A weakness is that while Koopman operator theory is stated as the inspiration of the architecture, the frequency-domain gating is not motivated from its principles. Therefore it is not clear what the final model is implementing in the perspective of Koopman theory.\n\nConsidering both the practical benefits and weaknesses in the theoretical analysis, I updated my score accordingly.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## Longer test-time forecast horizon\nWe have now conducted experiments for increased test-time horizon. Please see [Tab. 1](https://anonymous.4open.science/r/SKOLR-1D6F/Tab1_ScaleUp.pdf). SKOLR has a recursive structure. Even if we train over a given horizon, we can recursively predict for a longer horizon. There is performance deterioration, but our results show it is not severe. SKOLR's performance compares favorably with Koopa's.\n## Non-overlapping patch tokenization\nWe apply patch processing before Linear RNN branches. This reduces complexity to O(L/P) from timestamps (O(L)). Our efficiency results *do* use this for baselines (PatchTST, iTransformer, Koopa). It slightly improves baseline performance. \n## Hyperparameter selection + sensitivity\nWe will clearly specify the hyperparameter selection protocol in App.A.2. We selected the optimal configuration based on MSE for the validation split. Our setup strictly separates training, validation, and test sets, ensuring no information leakage.\nTab. 4 (paper) analyzes how branch number N and dimension D impact performance. Response [Tab. 3](https://anonymous.4open.science/r/SKOLR-1D6F/Tab3_PatchLen.pdf) provides new results of sensitivity to patch length. SKOLR exhibits little sensitivity to patch length P. We do not tune this in our experiments; we used P = L/6 for all datasets.\n## Revise Tab. 8\nWe apologize for inconsistencies in Tab.8. We've verified all results against our original records and fixed the MASE metric issues for the Quarter and Others categories in the revised [Tab.8](https://anonymous.4open.science/r/SKOLR-1D6F/Tab8_ShortTerm.pdf).\n## Relationship to [1] Orvieto et al. 2023\nThank you for highlighting this paper. We will modify the introduction: \"In this work, we consider time-series forecasting, and establish a connection between Koopman operator approximation and linear RNNs, building on the observation made by [1]. We make a more explicit connection and devise an architecture that is a more direct match.\"\n\nIn Related Work, we will add:\n\"Orieto et al. made the observation, based on a qualitative discussion, that the Koopman operator representation of a dynamical system can be implemented by combining a wide MLP and a linear RNN. We provide a more explicit connection, providing equations to show a direct analogy between a structured approximation of a Koopman operator and an architecture comprised of an MLP encoder plus a linear RNN. Although [1] observe this connection, their studied architecture stacks linear recurrent units interspersed with non-linear activations or MLPs. While excellent for long-range reasoning tasks, this departs from the architecture in their App. E. By contrast, our developed architecture does consist of (multiple branches) of an MLP encoder, a single-layer linear RNN, and an MLP decoder. It thus adheres exactly to the demonstrated analogy between Eq. (5) and (8) of our paper. Whereas our focus is time series forecasting, [1] target long-range reasoning. Although it is possible to convert their architecture to address forecasting, performance suffers because it is not the design goal.\"\n\nWe recognize the importance of acknowledging [1], but we don't believe that its existence significantly diminishes our contribution. The connection observed by Orieto et al. is qualitative; there are no supporting equations. In contrast, we develop an explicit connection by expressing a linear RNN in (5) and a structured Koopman operator approximation in (8). This explicit connection adds value beyond the qualitative insights.\n## Presentation - Sec. 3\nWe apologize for notational confusion and will correct it. There are strong motivations for the frequency decomposition design choice. In classical time-frequency analysis, the value of adaptation to different frequencies has long been recognized. Wavelet analysis applies different filters at different frequency scales. In more recent forecasting literature, frequency decomposition has been shown to be highly effective in TimesNet (Wu 2023), Koopa (Liu 2023), and MTST (Zhang 2024). Low-frequency dynamics may be considerably different from high-frequency dynamics and are more easily learnt after disentanglement. We are motivated to allow for observable functions to be learned both in frequency and time, with explicit consideration of the frequency aspect.\n\nWe will modify Sec.3 to describe the whole pipeline for a single branch, and then describe the multiple-branch case.\n## Questions: Non-linear systems; Fig. label\nTab. 1 signals are real system measurements, e.g., Electricity Transformer Temperature. We do not know the exact dynamics. The 4 systems in Sec.4.2 are commonly-studied, synthetic non-linear systems. These allow us to study a setting where it is important to model non-linear dynamics. We will write \"Synthetic Non-linear Systems\" to stress the synthetic nature.\n\nThe left figure of Fig.3 corresponds to (a) and the right to (b). We will add clear labels.", "claims_and_evidence": "The paper is focused on demonstrating state-of-the-art performance of the proposed method, and the main results are given in Table 1 and Figure 2, which supports the main claim. Also, the claim in page 5 that multi-branch parameterization reduces the parameter count is supported by Section 4.3.1. On the other hand, limitations of the presented evidences are as follows:\n\n- The authors have not shown whether it is possible to use a longer forecast horizon at the test-time, as in Section 5.3 of Liu et al., (2023). The utility of the method would be more convincing if this capability is demonstrated, especially given that both Koopa and SKOLR are based on Koopman operator theory.\n- In Appendix A.2, the authors claim they adopt non-overlapping patch tokenization after trainable gating in frequency domain. It is not clear how much this impacts the model efficiency compared to not tokenizing. Also, it is unclear whether this can be applied to other baselines (e.g., Koopa) as well. For fair comparison of efficiency one would expect patch tokenization to be consistently applied (or not applied) to all methods.\n- The hyperparameter selection protocol in Appendix A.2 is not very clearly described. Is the validation split used for the grid search? Also, having some analysis on the sensitivity to the hyperparameter choice (e.g., Appendix C of Liu et al., (2023)) would make the results more convincing.\n- In Table 8, some of the numbers are also not consistent with the reported results in Liu et al., (2023), and the highlightings of the MASE metric for Quarter and Others are wrong.\n\nLiu et al. Koopa: Learning Non-stationary Time Series Dynamics with Koopman Predictors (2023)", "methods_and_evaluation_criteria": "The evaluation criteria closely follows a prior work (Liu et al., (2023)) which seems sound to me.", "theoretical_claims": "The paper does not make theoretical claims.", "experimental_designs_or_analyses": "Please see the above sections.", "relation_to_broader_scientific_literature": "The paper is related to the fields of linear RNNs, time-series forecasting, and Koopman operator theory. Specifically, the contributions are in the application of Koopman operator theory for designing a linear RNN for time-series forecasting.", "essential_references_not_discussed": "The observation that Koopman operator theory can be connected to linear RNNs is not new, for example we have Orvieto et al. Resurrecting Recurrent Neural Networks for Long Sequences (2023), specifically Appendix E.1.2.", "other_strengths_and_weaknesses": "Strengths\n- The empirical results are promising, especially considering the computational efficiency of the proposed method (but see the concerns in the Claims And Evidence section).\n\nWeaknesses\n- The presentation of the methodology in Section 3 could be overall improved. Especially, while the authors have invested a significant amount of text describing Koopman operator theory, linear RNNs, and EDMD, the design of the encoder which uses trainable frequency-domain gating is not motivated concretely from the background theory and seems rather arbitrary. This is a key design choice with its own ablation study, but I do not see how this is motivated. Also, the mix-up of symbols (e.g., using $\\mathbf{Y}$ both before and after the frequency-domain gating) makes the description quite confusing.", "other_comments_or_suggestions": "For the presentation in Section 3, I suggest that the authors first describe the whole pipeline (including the encoder and decoder) for a single branch ($N=1$), and then describe the multiple-branch case. This is what is often done in sequence modeling papers using multi-head architectures.", "questions_for_authors": "- In Section 4.2, are there particular reasons to refer to these 4 systems as non-linear systems, given that the dynamics in Table 1 are also non-linear?\n- In Figure 3, what parts of the figure do (a) and (b) correspond to?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "jxJD0aSTq7", "reviewer": "Reviewer_hUoy", "summary": "This paper introduces a novel approach that connects Koopman operator approximations with RNNs for efficient time-series modeling. Koopman operator theory provides a linear representation of nonlinear dynamical systems but is typically infinite-dimensional, making it challenging to apply directly. To address this, the authors propose a structured approximation of the Koopman operator and establish its equivalence to linear RNN updates. Based on this insight, they develop SKOLR, a model that integrates learnable spectral decomposition of input signals with multilayer perceptrons (MLPs) as measurement functions and a highly parallelized linear RNN stack to implement the structured Koopman operator.\n\n## update after rebuttal\nSorry for the last-minute response. The authors have addressed most of my concerns. I have updated my score to \"weak accept\" accordingly.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## 1. High Efficiency\n### 1.1 Computational efficiency:\nWe provide additional results for all datasets. Please see [Tab.4](https://anonymous.4open.science/r/SKOLR-1D6F/Tab4_Computation.pdf). SKOLR achieves a compelling trade-off between memory, computation time, and accuracy.\n\n### 1.2 Theoretical Complexity Analysis\nSKOLR achieves computational efficiency through structured design and linear operations. For a time series (length L, patch length P, embedding dimension D, N branches):\n\n- **Time complexity**: O(N × (L/P) × D²) from spectral decomposition, encoder/decoder MLPs, and linear RNN computation\n- **Memory complexity**: O(N × D²) for parameters and O(N × (L/P) × D) for activations\n\nCompared to a non-structured model with dimension D' = N×D:\n- Non-structured approach: O((L/P) × N²D²) time and O(N²D²) memory\n- SKOLR provides N-fold reduction in computational requirements\n\nSKOLR avoids quadratic scaling with sequence length seen in transformers (O((L/P)² × D + (L/P) × D²) time, O((L/P)² + (L/P) × D) memory).\n\n### 1.3 Parallel computing\nThe N separate branches are processed independently (in our code), reducing time complexity to O((L/P) × D²). \n\nThe linear RNN computation has no activation functions, so the hidden state evolution is: $h_k = g(y_k) + \\sum_{s=1}^{L/P} W^s g(y_{k-s})$. This allows efficient matrix operations, reducing time complexity to O($D^3 \\log(L/P)$ + $(L/P)^2 \\times D$) per branch. For time series where $L/P \\ll D$, this is a significant speedup. \n\n## 2. Exceptional Performance Claim\nOur claims are based on Tab.s 1 \\& 2 and Fig.s 2 \\& 4. SKOLR requires less-than-half the memory of any baseline and less-than-half of Koopa's training time (Fig.4). SKOLR has the (equal-)lowest MSE in 17 our of 32 tasks and ranks second in a further 7 (Tab.1). SKOLR significantly outperforms Koopa for synthetic non-linear systems (Tab.2). Given the very low memory footprint, the low training time, and the impressive accuracy, we consider that the claim of \"exceptional\" performance is supported, but we can use a less strong adjective.\n\nWe agree that the paper would be strengthened by more examples (showcases) demonstrating the capture of complex patterns. We do have one example in Fig. 3, but we will include more. Please see [Fig. 2](https://anonymous.4open.science/r/SKOLR-1D6F/Fig2_Compare.pdf) as an example. SKOLR's predictions have much lower variance than Koopa's and track the oscillatory behaviour better.\n\n## 3. Relevant Reference\nWe will cite it and add discussion. The paper evaluates the existing non-linear RNNs and Koopman methods for near-wall turbulence. It does not draw connections or develop a new method. \n\n## 4. Clarity\n### 4.1 Frequency Decomposition\nWe agree that the motivation for our learnable frequency decomposition could be better. Please see the response to Reviewer kXAS. Learnable frequency decomposition offers three key advantages. (1) Each branch can focus on specific frequency bands, decomposing complex dynamics. (2) Learnable components adaptively determine which frequencies are most informative for prediction. (3) This approach aligns with Koopman theory, as different frequency components often correspond to different Koopman modes.\n\n### 4.2 Generalization capability\nConstructing theoretical guarantees that the approach generalizes is challenging and would be a paper all on its own. We do provide experimental results for a large variety of time-series that exhibit very different characteristics, ranging from strongly seasonal temperature series to pendulums exhibiting highly non-linear dynamics. We consider that the experimental evidence in the paper is strongly supportive of a capability to generalize to a diverse range of time series.\n\n## 5. Theoretical analysis: eigenvalues\nThis is an excellent suggestion. Our focus is forecasting, so our results and analysis concentrate on that task. However, the analysis of learned Koopman operator eigenvalues can indeed reveal important characteristics.\n\nWe analyzed eigenvalue plots for Traffic dataset (see [Fig.3](https://anonymous.4open.science/r/SKOLR-1D6F/Fig3_Eigenvalue.pdf)). We see that each branch learns complementary spectral properties, with all eigenvalues within the unit circle, indicating stable dynamics. Branch 1 shows concentration at magnitude 0.4, while Branch 2 exhibits a more uniform distribution. The presence of larger magnitudes (0.7-0.9) indicates capture of longer-term patterns. \n\n## 6. Questions\n### 6.1\nIt is a typo. We will revise Fig. 1 to show the proper time indexing across all components.\n\n### 6.2 Long horizon prediction\nTo show SKOLR's capability for longer horizons, we included experiments with extended prediction horizons (T=336, 720) in App. B.2 (Tab. 9), SKOLR maintains its performance advantage at extended horizons, with lower error growth rates.\n\nPlease also see response to reviewer kXAS concerning test-time horizon extension and [Tab. 1](https://anonymous.4open.science/r/SKOLR-1D6F/Tab1_ScaleUp.pdf).", "claims_and_evidence": "The claims in the paper are generally well-supported by clear and convincing evidence. However, there are still some problematic claims as follows:\n\n* High efficiency: the authors claim in there paper that the proposed architecture is computationally effective. However, they only provide experiment results in ETTm2 and Traffic datasets (in Fig. 5). The authors should include results of other datasets. Moreover, including theoretical analysis of computation complexity could strengthen this claim. The authors also mention that the branch decomposition make the linear RNN chains highly parallel. I suggest that the authors should also explain how they implement the parallel computing.\n* Exceptional performance: the authors claim that this architure delivers exceptional performance. However, I do not showcases in the paper. The authors should also present showcases to demenstrate that their model are effective in capturing complex temporal patterns (see line 366).", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria in the paper are well-suited for the problem. The authors bridge a gap between Koopa Operator Theory and RNN to achieve both efficiency and performance. The evaluation criteria (ECL, Traffic and other benchmarks & metrics) is widely used in time series forecasting evaluation.", "theoretical_claims": "I have checked the correctness of proofs for theoretical claims. If authors could also include theoretical analysis for computational complexity (discussed above), it would be better.", "experimental_designs_or_analyses": "I have checked the soundness/validity of experimental designs and analyses. The authors provide results in widely used time series forecasting benchmarks and nonlinear dynamical system and claim performance of the model and provide efficiency comparision in ETTm2 and Traffic to claim the efficiency. As discussed above, more showcases should be included and efficiency promotion should be analyzed in more datasets.", "relation_to_broader_scientific_literature": "The key contributions of this paper are related to the application of physics informed methods (Koopman Operator Theory in this paper) in time series. The proposed decomposition and RNN stack improve computational efficiency and accuracy compared to previous Koopman-based methods ([1] [2]).\n\n[1] Koopa: Learning Non-stationary Time Series Dynamics with Koopman Predictors.\n\n[2] Koopman neural forecaster for time series with temporal distribution shifts.", "essential_references_not_discussed": "There is some reseearches incorporating Koopman Theory into RNNs and the authors should briefly discuss the difference and novelty.\n\n[1] Recurrent neural networks and Koopman-based frameworks for temporal predictions in a low-order model of turbulence", "other_strengths_and_weaknesses": "Strengths: \n\n* Writting: This paper is well written and the figures could express authors' idea clearly. After reading this paper, I could easily understand the authors' idea. The work is well written providing sufficient relevant background knowledge on Koopman operator theory also for reader who is unfamiliar with this theory.\n* Strong performance in benchmarks: SKOLR outperforms or matches state-of-the-art models on multiple standard forecasting datasets. The model also demonstrates superior performance in nonlinear dynamical system modeling (e.g., Pendulum, Duffing, Lotka-Volterra, Lorenz ’63), validating its effectiveness beyond traditional forecasting tasks.\n\nWeaknesses: \n\n* Clarity: While the learnable frequency decomposition improves performance, it is not clearly justified why this approach is superior to other feature extraction methods. The paper lacks an in-depth exploration of why the Koopman-based linear RNN structure generalizes well to different datasets beyond empirical results.\n* Theoretical analysis: the essence of the deep Koopman method lies in the spectrum of the Koopman operator, because the eigenvalues determine the model's behavior during long-term evolution. However, the author did not provide any analysis or visualized results regarding the eigenvalues.", "other_comments_or_suggestions": "I am willing to raise my score if the authors could address my question.", "questions_for_authors": "1. I noticed in Fig. 1, the Encoder only untilizes $x_1$ to derive $z_i$ for $i=1,2...L$, is it a typo?\n2. Will this architecture suffer from error accumulation which usually happens in RNNs? How to solve this problem? Could authors provide some showcases for long context prediction?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0bTa9b7Ubi", "reviewer": "Reviewer_fgpm", "summary": "The authors develop an approach to forecast time-series, via Koopman operator theory, through the use of linear RNNs. The authors demonstrate that their approach delivers state-of-the-art performance on long and short term forecasting benchmarks and is significantly less computationally expensive.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## Strengthen Claims with more Convincing Evidence\n\n### Branch Decomposition - frequency band specialization\nWe appreciate the reviewer’s insightful feedback and agree that \"specialization\" is too strong a claim. We consider that Fig. 3 provides evidence that different branches place more emphasis on some frequency bands (e.g., Branch 1 has a greater relative emphasis on the low-frequency component). But there is not evidence of specialization. \n \n### ILI performance\nRegarding the performance of SKOLR on ILI, we recognize the concern about the phrasing of \"remarkable forecasting capability.\" \nWhile SKOLR achieves the best MSE among the tested models, we agree that this is mischaracterization, given that the error is still high. We will change the text to more accurately state that SKOLR outperforms the baseline methods, with significant error reduction for the shorter horizons.\n\n\n### Ablation Study - learnable frequencies\nThe observation on the ablation study is insightful. We agree with the general observation that the learnable decomposition is not a critical component and does not lead to a major improvement. On the other hand, it does lead to a relatively consistent 2-5\\% improvement, and it is not many parameters to learn. We will modify the text to downplay the significance of the learnable frequency decomposition, and explain that while it does bring some benefit, SKOLR performs well without it. It suggests that the key aspect of the multiple branches is to allow for MLPs to learn different embeddings.\n\n## Theoretical Claims - linkage between Koopman operator and RNN\n\nOur claim is not as strong as \"the Koopman representation can be equivalent to a linear RNN\". In the abstract and introduction, we were careful to write that we \"establish a connection between Koopman operator approximation and linear RNNs\". In Section 3.3, after equation 8, we write that \"this structure can be implemented as a linear RNN\". **So, our conclusion/claim is more that a linear RNN (with MLP embeddings) provides an efficient and effective mechanism for implementing a specific structured Koopman operator approximation (involving an augmented state incorporating past observations).** Our equivalence specifically refers to the scenario where we construct an extended state representation using lagged observations (Section 3.3), which effectively transforms the Koopman operator approximation into a form where Eq. 5 and Eq. 8 become structurally identical. \n\nSince your review is thorough, indicating careful attention to the paper, it seems that we haven't made it clear enough that our theoretical claim (which is indeed more of an algorithmic development) is more limited. We will modify the language to make it very clear. This is perhaps also an issue we would raise with the qualitative connection in Orvieto et al. (2023) between Koopman operator analysis and an MLP + linear RNN; when you write equations, the connection is not quite so simple, nor as general, as suggested in that work.\n\n\n\n## References\nThank you very much for highlighting these essential references. We will incorporate them to properly situate SKOLR within the Koopman literature. We'll add Mezic (2005) alongside Koopman (1931) as foundational work on Koopman mode decomposition, and Rowley et al. (2009) when discussing DMD's relationship to Koopman theory. We'll discuss connections between our spectral decomposition approach and Kutz et al. (2015), whose frequency-based decomposition shares similarities with our branch structure. Importantly, we'll acknowledge Arbabi and Mezic (2017) when discussing time-delay embeddings in Section 3.3, as their Hankel DMD provides theoretical support for our extended state construction. We'll also correctly attribute EDMD to Williams et al. (2015) in Section 3.2, while maintaining the Li et al. (2017) reference for neural network extensions. These additions will provide proper attribution and better contextualize our contributions within the evolving landscape of Koopman-based methods.\n\n\n## Weaknesses\n\nThe three weaknesses are addressed in the comments above.\n\n## Suggestions\n\nThank you for these careful observations. We will standardize our usage of \"time series\" (without hyphen) throughout the paper for consistency. \n\nWe also appreciate you flagging the timeline in our literature review. We will revise this section to include pioneering works such as [1]-[3] and properly reflect the development of time series forecasting methods.\n\n[1] Hochreiter, S., \\& Schmidhuber, J. (1997). Long short-term memory. Neural Computation, 9(8), 1735-1780.\n[2] Chung, J., Gulcehre, C., Cho, K., & Bengio, Y. (2014). Empirical evaluation of gated recurrent neural networks on sequence modeling.\n[3] Binkowski, M., Marti, G., & Donnat, P. (2018, July). Autoregressive convolutional neural networks for asynchronous time series.", "claims_and_evidence": "There are a few claims that I believe could be strengthened with more convincing evidence/change of language:\n\n1. That the different branches of SKOLR \"specialize in distinct frequency bands\" (page 6). Looking at Fig. 3 (left column), it is not particularly clear to me that the two branches \"specialize in distinct frequencies\". If anything, it looks like they agree on many of the frequencies. Clarifying this (and/or providing a quantitative comparison to make their point more clear) would be good. \n\n2. The performance of SKOLR on ILI is the best of all the methods tested. However, saying \"SKOLR demonstrates remarkable forecasting capability\", when the MSE (~1.5) is of a similar magnitude as the average ILI across a given flu season (e.g., the national baseline ILI for the 2018-2019 season was 2.2), is perhaps a little misleading. Toning down this remark is, in my opinion, more appropriate. \n\n3. In the ablation study (setting the learnable weights to 1) the results appear relatively consistent with the non-ablated model (the difference between the MSE is often different by <5%). This combined with the first point above (#1) makes me somewhat skeptical that weighting the branches differently really affects the performance significantly. Doing the same analysis on other data sets where perhaps it makes a bigger difference or changing the language to be more transparent that the weighting doesn't often lead to a large difference would be good.", "methods_and_evaluation_criteria": "The authors did a great job evaluating their approach. They thoroughly compared on baselines and provided evaluation of model size and computational resources used.", "theoretical_claims": "The main theoretical claim made by this paper (which is perhaps too strong of a word, as the authors more argue that two things are similar) is that the Koopman representation can be equivalent to a linear RNN. This inspires their branched decomposition. While I agree that Eq. 5 and Eq. 8 are the same, I think this claim is somewhat misleading. In particular, RNNs typically take in an input at each time step (the $U v_k$ in Eq. 4), in addition to propagating forward the internal state. This makes linear RNNs a dynamical system with control (or inputs). In Eq. 8, the Koopman operator is solely propagating forward the observable functions $g$. This is more equivalent to inputing some state into a linear RNN and then evolving it forward (without providing any more inputs). So while I think the equivalence is correct, it relies on viewing RNNs differently than they are typically thought of. Making this more clear is important.", "experimental_designs_or_analyses": "The experimental design and analysis all appears sound.", "relation_to_broader_scientific_literature": "The contributions of this paper are relatively well situated within the broader scientific literature. In particular, by comparing their method to Koopa, the current SOTA, they convincingly show some of the benefits of their approach. Discussing, in more detail, the advantage of having a smaller, less computationally demanding model (in the Introduction) could further strengthen their paper.", "essential_references_not_discussed": "Given that this area of using Koopman + ML for forecasting has exploded over the past few years, it is reasonable that there may be papers that the authors did not cite that I believe could be relevant. However, there are several foundational papers that the authors did not cite. Adding these (and discussing how the author's work relates) is important and will strengthen the paper. \n\n1. Mezic 2005 Nonlinear Dynamics - This work is the most relevant paper for discussing Koopman mode decomposition and the Koopman representation (and should be cited when citing Koopman 1931). \n2. Rowley et al. 2009 Journal of Fluid Mechanics - This was the first work to demonstrate that DMD could approximate the Koopman mode decomposition. \n3. Kutz et al. 2015 SIAM J. Appl. Dyn. Sys. - This work proposes a similar decomposition of data via FFT and I believe it employs a similar block structure as is used in SKOLR. \n4. Arbabi and Mezic 2017 SIAM J. Appl. Dyn. Sys. - This work was the first to rigorously propose using time-delays for constructing a Koopman operator (Hankel DMD). \n\nAlso, minor point, but the authors reference Li et al. (2017) when talking about EDMD (Sec. 3.2), but the first paper to develop EDMD was Williams et al. (2015) (which the authors do cite earlier and later).", "other_strengths_and_weaknesses": "**Strengths**\n1. This paper achieves SOTA performance (even if the gains are at times relatively small). The comparison with other methods was convincing. \n2. The improvement in decreasing model size was significant and is an exciting outcome of the work. \n3. Fig. 1 was helpful for understanding the method. \n4. Comparing how the use of different number of branches affected performance (Fig. 4, Tables 3 and 4) was interesting.\n\n**Weaknesses**\n1. As noted above, there are several claims that I think need to be clarified/strengthened. \n2. As noted above, there are several important papers that need to be cited and discussed. \n3. As noted above, clarity is need when discussing the \"equivalence\" with the linear RNNs.", "other_comments_or_suggestions": "**Minor points**\n1. (very minor) The authors write \"time series\" throughout the paper, but the title has \"time-series\" \n2. (very minor) The authors write that \"early methods used RNNs and CNNs\" (page 8), but then cite papers from 2020/2024.", "questions_for_authors": "I do not have any other questions for the authors.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "yesiQWRXPE", "reviewer": "Reviewer_qBxE", "summary": "This paper introduces SKOLR, a structured Koopman operator-based approach to time-series forecasting that connects Koopman operator approximations with linear recurrent neural networks (RNNs). By representing dynamic states using time-delayed observations, the method approximates the Koopman operator in a structured manner through a parallel linear RNN stack. The proposed approach integrates a learnable spectral decomposition of the input signal with a multilayer perceptron (MLP) as measurement functions. SKOLR is evaluated on standard forecasting benchmarks and nonlinear dynamical systems, where it appears to achieve competitive results.", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## Claim 1 \nWe would kindly ask the reviewer for more supporting arguments for the assertions. \n\n> \"connection is somewhat superficial\" \n\nWe establish a clear connection between an EDMD-style approximation of the Koopman operator for a history-augmented state and a linear RNN. This motivates a highly effective but simple architecture, which turns out to be not superficial.\n\n> \"theoretical justification is not fully developed\"\n\nWe contend that we have fully developed the theoretical justification mathematically.\nEq. 5 is the equation for a linear RNN with a finite history. Eq. 6-7 present the EDMD-style approximation to the Koopman operator. And Sec. 3.3 explains that if we consider an augmented state with historical observations then we can rewrite the approximate Koopman operator relation as Eq. 8. Obviously, Eq. 8 has the identical structural form as Eq. 5, implying that we can implement Eq. 8 using a linear RNN (with an MLP embedding). \n\n> \"only intuition is provided\"\n\nThe development is by no means \"only intuition\". It is a clear and concise mathematical development.\nHowever, we are happy to provide a more formal proposition-proof structure in an appendix if this is desired.\nPlease flag this, and we will provide it in the final response.  \n\n> \"assumptions ... are not explicitly stated\"\n\nWe kindly ask which assumptions are missing in this development? The key assumptions are expressed in the fact that we are applying EDMD-style approximation (so the EDMD assumptions apply); we specify the full rank assumption for a unique solution.\n\n> \"a very similar equivalence was proved already\" \n\nApp. E [a] contains NO equations connecting linear RNNs to Koopman theory — only a **qualitative observation (\"Sounds familiar? This is exactly what a wide MLP + Linear RNN can do.\")** without mathematical formulation or proof. While we acknowledge this prior insight, our contribution provides the explicit mathematical connection that was not previously established.\n\n## Connection to [a] and [b]\nWe would include the suggested works into our discussion.\n\nHowever, we'd like to highlight key distinctions from [a]. As aforementioned, [a] only offers a qualitative discussion of Koopman operator theory relevance, listing equations for Koopman theory (Appx. E, Eq.s 17-22) followed by a conclusion without any mathematical development. \nMore importantly, the architecture in [a] does not follow the qualitative observation in Appex. E: the stack of [linear recurrent units + MLPs] has strayed away from the Koopman operators. \nIn contrast, our architecture is an exact match to our derivations in (5) and (8).\n\nFurthermore, [a] does not address time-series forecasting with any experiments on this task. In fact, the proposed architecture in [a] performs poorly in the time-series forecasting setting (See [Tab. 7](https://anonymous.4open.science/r/SKOLR-1D6F/Tab7_LRU.pdf)).\n\nConcerning [b], we will cite it and include more discussion about the intersection of machine learning methods and Koopman operators.\n\n## Statistical Significance\nTo address the reviewer's concern about statistical significance, we add [Tab. 6](https://anonymous.4open.science/r/SKOLR-1D6F/Tab6_Std.pdf) with standard deviation (std) across 3 independent runs for all datasets and forecast horizons. The low stds (<0.003 for most datasets) demonstrate the consistency and reliability of SKOLR's performance.\nThus, we contend that SKOLR's improvements, even when small in magnitude, represent genuine performance differences rather than statistical fluctuations. \nAdditionally, we will provide confidence intervals and statistical significance tests in our final response.\n\n## Efficiency and Parameter Reduction\nWe will add a dedicated \"Theoretical Complexity Analysis\" section to formally establish SKOLR's computational benefits. Due to the word limit, please refer to response to reviewer hUoy (point 1.2) for detailed information.\nThis comprehensive theoretical analysis, combined with our expanded empirical results in [Tab.4](https://anonymous.4open.science/r/SKOLR-1D6F/Tab4_Computation.pdf), provides thorough justification for SKOLR's efficiency claims.\n\n## Ablation study\nAs suggested by the reviewer, we have also conducted a more comprehensive ablation study on the design elements of SKOLR.\nAs shown in [Tab. 5](https://anonymous.4open.science/r/SKOLR-1D6F/Tab5_Ablation.pdf), we compare our full SKOLR model with two ablated variants: (1) \"w/o Structure\": no structured decomposition, using a single branch with dimension (N×D); (2) \"w/o Spectral Encoder\": no learnable frequency decomposition, while maintaining the multi-branch structure.\n\nThe results show that both components contribute meaningfully. Removing the structured decomposition leads to performance degradation on 27/32 tasks, with notable declines on ETTh1 and ILI, while increasing computational overhead. Similarly, removing spectral encoder impacts performance on 23/32 tasks, though with a smaller overall effect.", "claims_and_evidence": "The paper makes several key claims:\n1. Establishing a Connection Between Koopman Operator Theory and Linear RNNs – The authors show an equivalence between Koopman operator approximations and linear RNN updates, but this connection is somewhat superficial. First, this is not a strictly novel result, as a very similar connection was shown in [a] (see appendix E). Furthermore, I think that theoretical justification is not fully developed, and the assumptions required for this equivalence to hold are not explicitly stated.\n\n2. Superior Forecasting Performance – While the reported numbers show competitive results, the claim of superiority cannot be convincingly ascertained. Many of the performance gains are on the third significant digit, and since variances are not reported, the differences between methods might be due to statistical uncertainty. \n\n3. Efficiency and Parameter Reduction – The authors show that SKOLR achieves competitive results with reduced computational costs.\n\nOverall, the claims lack sufficient theoretical and empirical justification, and the evidence presented does not unambiguously support SKOLR’s superiority.\n\n[a] \"Resurrecting Recurrent Neural Networks for Long Sequences\" by Orvieto et al. 2023", "methods_and_evaluation_criteria": "The evaluation criteria largely follows [b], and tests SKOLR on standard time-series forecasting benchmarks. As I already mentioned, a big shortcoming of the evaluation is the lack of reported standard deviations. As many of the baselines attain similar errors, single estimates of the mean absolute error are, in my opinion, not sufficient to evaluate a method.\n\n[b] \"Koopa: Learning non- stationary time series dynamics with koopman predictors.\" by Liu et al. 2023", "theoretical_claims": "As mentioned above, the main theoretical claim of the paper is the equivalence between linear RNNs and Koopman operator theory. For this equivalence, only intuition is provided, there is no formal theorem or clear set of assumptions under which the approximation holds. Furthermore, a very similar equivalence was proved already in [a]. \n\n[a] \"Resurrecting Recurrent Neural Networks for Long Sequences\" by Orvieto et al. 2023", "experimental_designs_or_analyses": "I think that the analysis of the method is far from comprehensive. Indeed, SKOLR is an assembly of many inter-dependent components:\n1. Spectral encoder and spectral decoder\n2. Structured RNN stack\nWhile an \"Ablation Study\" is presented in Section 4.3 the Authors mainly test how different settings in the encoder and RNN stacks impact the final performance. I would classify it as a study of the dependence on hyperparameters. An ablation study, conversely, should selectively remove components (such as the spectral encoder, or the structured decomposition of the RNN) and see the impact on performances.\n\nWhat Section 4.3 shows, instead, is that if the hyperparameters of SKOLR are not well-tuned, the claimed performance gains are completely washed away.", "relation_to_broader_scientific_literature": "This paper does not sufficiently engage with prior work. Koopman-based learning is an active research area, yet the paper misses key references on data-driven Koopman approximations and alternative learning-based Koopman methods. See the essential references not discussed, below.", "essential_references_not_discussed": "For linear RNNs, please check the aforementioned [a], establishing the connection between Linear RNNs and koopman operators. More generally, there is a growing literature on machine learning methods for Koopman operators, see e.g. [b]\n\n\n[a] \"Resurrecting Recurrent Neural Networks for Long Sequences\" by Orvieto et al. 2023\n[b] \"Learning dynamical systems via Koopman operator regression in reproducing kernel Hilbert spaces\" by Kostic et al. 2022", "other_strengths_and_weaknesses": "Strengths:\n- The paper provides a method for time-series forecasting connecting Koopman theory with RNNs.\n- The empirical results show competitive performance, even if not definitively superior.\n- The method is computationally efficient compared to transformers.\n\nWeakneses\n- The theoretical claims are underdeveloped and were already presented in prior works, which the Authors do not acknowledge.\n- The experimental results do not convincingly demonstrate clear superiority over existing methods.", "other_comments_or_suggestions": "- More rigorous ablations should be included to demonstrate the necessity of each architectural component.\n- The paper should engage more deeply with prior literature to clarify its novelty.\n- More discussion on failure cases and limitations would improve transparency.", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: the paper motivates the architecture, shows strong forecasting results across 8 real-world benchmarks and 4 nonlinear dynamical systems, and lower parameter/time cost compared to Koopa and Transformer baselines. The author's rebuttal addressed reviewers' main concerns (frequency-band “specialization” claim, statistical significance, efficiency analysis, additional ablations).

Reviewer qBxE remains opposed, citing inadequate formalism in the Koopman–RNN equivalence and missing assumptions/notation. The rebuttal provided a line-by-line response, clarifying assumptions, adding a proposition-proof outline, expanded ablations, and variance statistics which I think confirms validity of the results. 

the qualitative link between Koopman operators and linear RNNs was hinted in prior work—but SKOLR turns this insight into a compact, easily parallelized architecture that attains consistently top or second-best accuracy with ~½ the memory of current SOTA. Given (i) broad interest in lightweight sequence models, (ii) solid empirical evidence with low variance, and (iii) responsiveness to reviewer feedback, the paper meets the bar for the conference.
Decision: Accept (poster)