Environment Inference for Learning Generalizable Dynamical System
Authors: Shixuan Liu
Yue He
Haotian Wang
Wenjing Yang
Yunfei Wang
Peng Cui
Zhong Liu
Venue: NeurIPS 2025 spotlight ()
Keywords: Dynamical Systems
Distribution Shift
Environment Inference
Primary area: machine_learning_for_sciences

Abstract: Data-driven methods offer efficient and robust solutions for analyzing complex dynamical systems but rely on the assumption of I.I.D. data, driving the development of generalization techniques for handling environmental differences. These techniques, however, are limited by their dependence on environment labels, which are often unavailable during training due to data acquisition challenges, privacy concerns, and environmental variability, particularly in large public datasets and privacy-sensitive domains. In response, we propose DynaInfer, a novel method that infers environment specifications by analyzing prediction errors from fixed neural networks within each training round, enabling environment assignments directly from data. We prove our algorithm effectively solves the alternating optimization problem in unlabeled scenarios and validate it through extensive experiments across diverse dynamical systems. Results show that DynaInfer outperforms existing environment assignment techniques, converges rapidly to true labels, and even achieves superior performance when environment labels are available.

Environment Inference for Learning Generalizable Dynamical System
Shixuan Liu
Yue He²* Haotian Wang' Wenjing Yang' Yunfei Wang
Peng Cui Zhong Liu
Cooleee of Computer Science and Technology. National University of Defense Technology
"School of Information, Renmin University of China
College of Systems Engineering. National University of Defense Technology
'Department of Computer Science and Technology, Tsinghua University
"Laboratory for Big Data and Decision, National University of Defense Technology
szftandy@hotmaill com, hy8658654gaail com
(wanghaotian13, wenj ingiiinngg..n. yang, wangyunfei, i Liuzhong) @nudt edu. cn cuip@tsinghua edw.g edu. en
Abstract
Data-driven methods offer efficient and robust solutions for analyzing complex dynamical systems but rely on the assumption of L.I.D data, driving the develop- ment of generalization techniques for handling environmental differences. These techniques, however, are limited by their dependence on environment labels, which are often unavailable during training due to data acquisition challenges, privacy concerns, and environmental variability, particularly in large public datasets and privacy-sensitive domains. In response, we propose Dynalnfer, a novel method that infers environment specifications by analyzing prediction errors from fixed neural networks within each training round, enabling environment assignments directly from data. We prove algorithm effectively solves the alternating optimization problem in unlabeled scenarios and validate it through extensive experiments across diverse dynamical systems. Results show that Dynalnfer outperforms existing envi- ronment assignment techniques, converges rapidly to true labels, and even achieves superior performance when environment labels available.
Introduction
Data-drived approaches, especially neural networks, offer a powerful alternative or complement to traditional physics-based methods for understanding complex dynamical systems 4. Neural network-basse emulators are particularly valuable for their ability to provide fast, cost-effective approximations of complex simulations (9 [22] making them especially useful in scenarios where the underlying physics are poorly understood or misinterpreted, or where external disturbances are difficult model 43 34 These emulators are adept at handling large sets of variables and solving problems that are challenging for conventional solvers. Recent advancements in deep learning, along with innovative methods for modeling temporal and spatio-temporal systems, have led to a significant increase in applications across various fields, ranging from simple Hamiltonian dynamics to more complex areas like fluid dynamics and climatology [1.
While recent advancements have shown promising results, they often rely on the assumption that abundant, static data are available to satisfy the independent and identically distributed (IID) hypothe- sis. However, this assumption is frequently violated in practice due to challenges in data collection, associated costs, and environmental changes driven by exogenous factors 24 26). Recent work
"Corresponding authors
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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in dynamical systems addresses this by introducing multi-ennvioommnt setting, where trajectories follow distinct dynamics across environments. These studies developed generalization methods that learn shared global component while accounting for environment-specific variations, avoiding the limitations of underperforming averaged models
Parameter Difference
Station
Station B
Required by DOD Methods to infer Similarities Differences
Trajectories
Env. Label A
Env. Label
From
B
Unavailable
Figure 1: Environment labels, required by current generalization methods, are often unavailable.
Nevertheless, a key limitation of many generalization techniques is their reliance on partitioning datasets across distinct domains or environments, which are assumed to capture underlying variations. These environment labels enable algorithms to identify and exploit both similarities and differences across environments. However, obtaining such environment labels during training is often challenging due to data acquisition difficulties or privacy constraints. For example, in scientific research, data may be collected over time under uncontrolled or unknown conditions 42). In ecological studies, critical environmental parameters such as temperature or rainfall may vary unpredictably or remain unrecorded. [2). Similarly, when aggregating data from multiple sources, environment labels frequently lost or omitted, a common issue in large public datasets (35). Furthermore, in privacy- sensitive domains like healthcare, finance, social networking, access environment-specifte information is often restricted (18). These limitations highlight the need for generalization methods that do not depend on explicit environment labels.
To address the challenge of unknown environment labels, we propose a novel approach that infers environment specifications by leveraging the key insight that trajectories within the same enviromment share consistent dynamics and exhibit similar prediction losses under the same neural network. This inherent consistency enables
automatically derive meaningful environment assignments directly from the training data. We introduce an environment inference objective designed for dynamical systems, which minimizes environment-sponn speecfii prediction losses. Using fixed neural networks, first infer environments and then iteratively refine these networks with the inferred environments, ultimately learning generalizable dynamical system.
Our model identifies environment labels directly from mixed trajectories of dynamical systems, facilitating the training of off- the-shelf generalization algorithms in scenarios where such labels are absent. Importantly, our findings demonstrate that inferring environments from mixed sequence data can improve the performance of generalization strategies, even compared to cases where environments are manually assigned.
Our main contributions are as follows:
We present the first investigation into the challenge of unlabeled environment conditions in the context of learning generalizable dynamical systems, and propose general framework named Dynalnfer that utilizes the prediction loss to accurately infer latent environment labels from mixed sequence data?
We theoretically establish that our algorithm effectively solves the alternating optimiza- tion problem without requiring environment labels, demonstrating its capacity to discern heterogeneous environments and infer generalizable mechanisms.
We examine the efficacy of Dynalnfer through experiments in both in-domain settings and adaptation scenarios using three representative dynamical systems. Results confirm that the environment labels assigned by DynaInfer converge rapidly to the true labels.
"Code available at http:////..ttttooooooo//aaaaaa/t

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The remainder of this paper is structured as follows. Section 2 clarifies the problem definition. Section introduces our framework and provides the theoretical underpinnings. Section 4 details the experimental setup and discusses the results. Related work is reviewed in Section 5, and Section 6 concludes the paper.
2 Problem Definition
2.1 Dynamical Systems
We examine dynamical systems determined by unidentified differential equations evolving over time, expressed as,
dæ dt
f(xz)
where E R is the time index within time interval [0, TT. and 2;, is a time-variant state within bounded set A. The evolution function f AA + T.A maps xy to its temporal derivative the tangent class of vector fields F.
space T-A and belongs
In this paper, we consider both ordinary differential equation (ODE) and partial differential equation (PDE). For ODEs, A C R for PDEs, A represents d' d'-deennsional vector field within bounded spatial domain (such 2D 3D Euclidean space) denoted S C Rd The function f characterizes the data distribution trajectories T. Trajectories initiated from To ~ p(Xo) are computed by
integrating the derivatives: 2₁ = zo + J f (zw)du, Vt E I.
2.2 Multi-Environment Dynamical Systems Learning
In contrast to the standard expected risk minimization (ERM) framework, which assumes i.i.d. trajectories, the multi-enviromment learning problem involves learning trajectories from M different environments. In each environment E [M] = {1, 2, .......,... M). J. the trajectories are governed by unique differential equations described by function fe. Specifically, consider N trajectories {u² ",, ). where each trajectory x² is associated with an environment e, E [M]. The dynamics of each trajectory x² thus modeled by the differential equation dx /dl = Se, (xr)). The
set of environments for all trajectories is denoted by e {e1, e2. eN} E [M]
In cenliiennvroommnt learning, the goal is to enhance traditional ERM methods by exploiting both the commonalities and disparities across diverse environments. To this end, the dynamics is decomposed into two components: global component shared across all environments, parameterized by 6. and an environment-speccffe component, parameterized by oe for each environment e. The set of enviroiment-specific parameters is denoted by & = { Consequently, the dynamics of each trajectory z' are parameterized by both the universal and environment-specific parameters,
dæt h 0z):0.00).
This parametrization entails a decomposition that can implemented either functionally or paramet- rically. The functional decomposition, expressed as h (æt; 0, Φ.) fo (x+) + go., (xt). distinguishes between a shared function fe and an environment-specific function Jo. 43 Alternatively, the parametric decomposition integrates the anvironment-speciff parameters directly, formulated as (11; 0,,00,d,, (b.) = fo+o (at) [7]. Intuitively, the key ingredient for multi-environment learning is that 0 should encapsulate the maximal shared dynamics, whereas áe should exclusively reflect the unique characteristics of each environment e not described by 0. However, directly optimizing both parameters poses ill-posed problem, often resulting in trivial solutions where the global component learns nothing meaningful. To counteract this, the regularization term (()e) is introduced to effectively penalize de, thereby facilitating learning in the global component. Consequently, with the information about the environments e (c1, C2,.... eN ). the loss function given by,
drf
M
Re +)
Σ 1=1
(at:0, den))
dl + Σ
M(d.).
(2)
di
IET
3

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The first term evaluates the regression precision of the parameterized function h(=; 8, 6c). The ground truth vector field (VF) is not explicitly known and derived from trajectory data. Using the learned VF, simulated trajectory is generated and used to calculate the regression loss by referring to real trajectories during training. The term (((e)) serves as a regularization term for Pc. with À controlling the intensity of the regularization.
2.3 Environment Inference for Multi-Environment Learning
In many real-world scenarios, the enviromment label for a trajectory sample is unknown. We aim infer an environment assignment for each sample that maximizes the model's generalization ability across different environments. To achieve this goal, reformulate the learning objective into an optimization problem contingent on a specific environment assignment e. Specifically, our aim is to learn the environment assignment {ê e² ..... e²' [M for each trajectory effectively optimize Equation The overall objective is defined as follows:
0¹, x² = arg min Rs (0, Φ).
20.
In this paper, explore particularly challenging scenario where the total number of training environments M also unknown. We investigate the development of a practical model that maintains favourable performance even when the exact number of true environments is unknown.
3 The Dynalnfer Framework
In this section, we introduce our framework that operates on field functions without prior domain knowledge, proving especially effective in dynamical systems where exogenous factors unob- served and in situations where relevant environmental information is unclear or absent. While some clustering methods infer labels for CV data, they operate miite-dimensional vectors in Euclidean space, which drastically differs from field functions, making them inapplicable.
The optimization challenge in Equation 3 is primarily due to the inherently discrete nature of the environment assignments é, which take values the set [M]. This discrete categorization impedes the direct application of traditional gradient descent methods, which are typically designed for continuous parameter spaces. To effectively address this challenge, we develop a dual iterative strategy that concurrently updates the environment assignments è and the model parameters 0, Φb The first step in our approach centers on inferring environment labels by analyzing the prediction errors of the trajectories output by the neural network during the current training round. This analysis serves as a diagnostic tool to uncover critical discrepancies that signify distinct dynamical environments. Following this, the second step entails refining the neural network parameters based the newly inferred environment assignments in an unbiased manner, enabling the neural network to precisely adapt the unique characteristics of each identified environment. Through this adaptive refinement, our model progressively enhances its accuracy and generalization capability across different dynamic settings. The complete method is detailed in Algorithm [ and is visually depicted in Figure 2
Algorithm Dynalnfer framework
Mixed
Trajectories
1: Input: Randomly initialized 0, & = (de hee assumed number of environments M, total rounds T,
Step 1: Infer
Fixed NNs(r
2: ((0), (0)
0, &
Environment
3: for f- 1-110 T, do
Assignment e
4: Update ((r) based on Equation 4
Step Optimize
5: Update g(r), (r) based on Equation
Update
NNs
6: end for
7: Output: 0, Φb
Figure 2: Model Framework.
3.1 Bias-aware Environment Assignment
The environment inference step receives a single dataset as input and generates a partition of the data into multiple environments. Intuitively, trajectories originating from the same environment adhere to

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consistent dynamics. Employing the same neural network to model these trajectories should yield similar estimation error across them, reflecting a coherence in their dynamic parameters.
Upon examination, we observe that the optimization framework defined in Equation (2 shares conceptual similarities with classical centroid-hased clustering methodologies, although the latter generally operate in Euclidean space. In K-means clustering, the primary goal is to minimize the within-cluster sum of squares, often referred as cluster inertia (37). This minimization effort concentrates on reducing the distances between the points within each cluster and their corresponding centroid, which typically converges to local optimum. This characteristic enables K-means to efficiently delineate distinct and compact clusters, capturing the core essence of data distribution with respect spatial proximity.
This insight prompts us to explore a conceptual analogy wherein the neural network that minimizes the loss most effectively operates analogously to "centroid" for cluster of trajectories within the same dynamic environment. We characterise the distance between a trajectory (data point) and the network (centroid) by the regression loss the trajectory using the network. Initially, with a randomly initialized Hwwork-analogour to randomly initialized centroid in K-means clustering we assign each trajectory label based on the minimal prediction loss calculated from all available networks. Subsequently, we refine this "centroid" by optimizing it to minimize the loss as specified in Equation (2. Through this iterative optimization process, we can achieve the objective stated in Equation (3).
More specifically, at round r, given the fixed network parameters from the previous iteration ((r-1), (((11), the environment assignment efTr is updated through the following process,
dx
h (ef:0(01)) 10000,0,6-1)) o6- dt.
arg min
JtEr
(E(M]
multiple solutions exist for Equation (4 and é e/r-1) minimizes it, we retain this assignment for the next round, i.e., (()) 6
()--1) This approach ensures the validity of a constant loss reduction (to be
stated in Proposition
3.2
Assignment-driven Optimization
After assigning trajectories to specific clusters, we proceed to update the conceptual centroid by optimizing network parameters. In the K-means algorithm, centroids recalculated by averaging the positions of all points within each cluster. Similarly, our method updates network parameters by considering the mean estimation error over trajectories within cluster, ensuring unbiased contributions from each trajectory. This approach not only improves the representational accuracy of each cluster but also enables the network dynamically adapt the underlying structure of the trajectories, thereby enhancing the efficacy and reliability of our learning process in unlabeled scenarios. Therefore, the parameters ((r) and (r) are given by:
g(r) (r) = arg min Rgin) d).
(5)
0.0
3.3 Theoretical Property
We begin by demonstrating that Algorithm AlgorismmDeffectivee optimizes Equation 3
Proposition 3.1. For all rounds S < Tr, we must have
Rai-i (g(++1) (0++))(((++)) (l(+1)
Rem. (0)))
Furthermore, suppose the space of arg ming, & Re (0, Φ) is finite for all è E [M] Then there exists a constant C' > 0 such that i > 1 and Rgiet (0(r+1), ((r+1)) < Racs (0(r), (r)),, we must have
Rao-413 ((0+1))000 (r(+1) Raco (pir) (6(1)) C.
Remark 3.1. Given the assumptions made in prior works 43 17 that (1; 0, d.)) is linear with respect to 0 and th., and that ((o..) is strictly convex with respect to d.. it follows logically that the space
of arg ming Ra (f, Φ)) is finite for all è E [M] as is evident from Equation 2 The proof is provided in Appendix
5

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This proposition demonstrates that, as long as the loss in consecutive rounds of Algorithm I decreases, the loss must decrease by constant C' 0.
4 Experiments
Our experiments investigate three dynamical systems governed by specific differential equations: an ODE for biological modeling, PDEs for reaction-diffusion in chemistry, and the Wavier-Stokes equations for incompressible fluid dynamics. These complex, nonlinear systems test our method's ability to classify spatio-temporal patterns and physical laws across diverse environments.
4.1 Environment Specification
We provide basic introduction to the datasets here, with detailed descriptions in Appendix E Lotka- Volterra (LV) [3 The system models the dynamics between prey-predator pair an ecosystem, captured by the following ODE:
dm/dt am Bmm, dn/ dt ômn - 7
where and n represent the population densities of the prey and predator, respectively, and B, 8, and ry are the interaction parameters between the two species.
Gray-Scott (GS) 28 The model uses simple reaction-diffusion equations to effectively study complex pattern formation chemical and biological systems, following underlying PDE dynamies:
0m/0tt Am mn. F(1 - mm),
0n/ôt An mn² (F k-y.
where m and represent the concentrations of two chemical components in the spatial domain S with periodic boundary conditions; Dm and Dn are their constant diffusion coefficients; and F and are the reaction parameters that govern the spatio-temporal dynamic patterns.
Navier-Stokes (NS) [22 The Navier-Stokes PDE describes the motion of viscous fluid substances:
dm 0t = ~n + vAm + E, Vu = 0
where n the velocity field, m V x n the vorticity. both n and m lie in a spatial domain S with periodic boundary conditions, is the viscosity (fixed at 1e-3), and is the constant forcing term in the domain S.
4.2 Experimental Setting and Baselines
Settings. We evaluate Dynalnfer in two distinct settings: in-domain generalization E, and adaptation to new environments in Eg, with & and E,, hosting disjoint environments, For in-domain experiments, both training and testing occur E.. At test time, environment labels are also not provided and are instead inferred from the prediction bias over an initial segment of the trajectory (less than 2 for practical reasons). For adaptation experiments, practice: initial training on the source domain & is followed by fine-tuning and testing on the target domain E,,, where environment labels are provided.
follow standard domain adaptation
Dataset Preparation. For in-domain experiments, we generate four LV trajectories each of nine environments, ten GS trajectories in each of three environments, and eight NS trajectories in each of four cnvironments. For adaptation experiments, we simulate the same number of trajectories per environment, conducting finetuning in two additional environments E E,, All dynamic environment parameters are detailed in Appendix B For evaluation, we sample 32 trajectories per environment, initialized according to the underlying distribution p(z,). The LV and GS data are generated using the DOPRI5 solver g while the NS data simulated with the pseudo-spectral method as in [2
We explore three potential strategies for assigning enviromment labels in the absence of environmental information, compared to our method (DynaInfer): grouping all samples into a single environment (All in One), assigning distinct environment label to each sample (One per Env), and random assignment (Random). Additionally, we consider an "Oracle" assignment method where labels are fully known during training, bringing the total to five labeling strategies. Furthermore.
Baselines.

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we consider three base models for dynamical system generalization: LEADS 4 CoDA-I1, and We utilize the neural network architectures and parameter configurations as described in their papers for each type of dynamic system. By combining these assignment methods with base models, we generate fifteen distinct methods for evaluation. In adaptation experiments, during fine-tuning, LEADS and CoDA adhere to the protocol described in their papers, by fixing the shared components or parameters and rendering only the Eu-specific components trainable. All neural network architectures, optimizers, and parameters for the base models are configured as described in their respective papers.
CoDA-12 [7].
Metrics. To rigorously evaluate predictive accuracy in dynamical system learning, we adopt two complementary metries: Mean Squared Error (MSE) and Mean Absolute Percentage Error (MAPE), averaged over 5 independent runs.
4.3
Experimental Results
4.3.1 In-domain Generalization Results
The in-domain generalization results detailed in Table Illlustrate the performance implications of various assignment strategies. We observe that the "All in One" and "Random" assignment strategies consistently underperform across multiple datasets and baseline models. While the "One per Env" strategy yields only mediocre results, it provides a viable initial approach in scenarios where no labels are available. Across all datasets, DynaInfer significantly outperforms other assignment strategies. Furthermore, DynaInfer consistently shows effectiveness across all tested base models and datasets, underscoring its robustness against diverse methods and datasets. Notably, Dynalnfer either matches or exceeds Oracle performance, particularly in complex PDE environments like GS and NS, suggesting that its bias-aware approach effectively compensates for not having access to the true labels available Oracle.
In Figure g DynaInfer's predicted states qualitatively align closely with the ground truth and Oracle, occasionally outperforming Oracle (e.g., in GS dataset with LEADS base model, where Oracle shows some jitters).
LEADS
CUDA-I
Dats
Assignment
Train
Test
Trato
Test
Trait
Tes
MSE
MSE
MAPE
All ia Oue
7:17B.
7.120002 22
49.2221.44
7.34
7
E2
49.4463.15
7.
7.41200002
39.20422.13
One
4.15 E-4
4911350
6.6822444
8-8
9.140
5.8751.01
8.8
84330.00 39 C-4
5.73&119
Random
B-30
7.2860022 B22
50.111.05
B-12
7.3980011
48.5721
8-2
7.39 E-2
48.8612
Dynalnter
4-4
7.932.499 E-5
283:1.22
9.57 K-5
1.8-3.00 E-4
3,2762.66
1.71 E-4
18223007 E-4
2/021166
7.0240.76
1,7800.10
3.190024
1,2680006
1.99
2.7220.88
1.3
All in Oze
8.73
4,000002 1
3008
502891.20
9.2 3
9612800 E3
4115
80123333
9
9.00
E
3725005713255
One Env
1.38
16680 54 B-3
173
(7344199.11
1.56
191:0.06 E-3
185 43663366143
B-2
1.870000 E:3
17408457.88 82
Random
8.78
9.3620
B
9.5950.22
8.7
9.35 E.
Dyualafe
3.60 E-5
4.1450.21 E-5
137.57533.00
9.22 K-5
1.2368.41
K-4
122.93222.0
6,99 E-5
7.25+2.11 R-5
112.52144155
Onucle
13340.76
07.77:12.09
6:
94063391
5
7.04
138.66166555
All in One
5.ML 12
6.7180111 12
259, 72214.8
3.99
66
E22
251 38133
6.17L
6.612000 L22
255 2049
Oue per Eav
22248-12
4.1100 E:2
169 88
345E-02
3.Hs0 E-2
16100110.73
2.31--02
4.04 E-2
2689.5
Random
02
6,286000
8.44
6.8
947.556339
5.78
6.66 E
254.4726.00
Dynalate:
6.108:04
7.0550.34 E:3
77.29:10118
1.235-02
1.220018 8:2
108.12110.30
K926-44
1.190012 B:2
96.57112.75
2.59E
6
67.581937
1.36E
19300.992
124.22612.35
7.11E
9.46 E
91.0625.8 85
Table 1: In-domain Experiment Results on the LV, GS, and NS environments. Our approach consistently outperforms all non-oracle assignment methods, and beats oracle at times, demonstrating its effectiveness in modeling heterogeneous environments and generalizing across dynamical systems.
4.3.2 Domain Adaptation Results
The results in Table 2 demonstrate the performance of different assignment strategies under the domain adaptation setting. The "One per Env" strategy consistently outperforms the "All in One" approach. While the "Random" " assignment benefits the base model LEADS, it slightly diminishes CoDA's performance across all datasets. Dynalnfer shows strong adaptation capabilities across various datasets and base generalization methods, consistently outperforming other non-Oracle techniques. This indicates that Dynalnfer effectively captures commonalities across environments, enabling smoother adaptation to new conditions. Furthermore, the performance gap between DynaInfer and the Oracle is significantly narrower in adaptation tasks compared to in-domain generalization.
7

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LEADS
CoDA
Oracle
Dynalnfer
Oracle
DynaInfer
Truth
Truth
Figure 3: Left: Predicted test trajectories from 3 environments vs. ground truth and Oracle for LV with base models. Middle and Right: Predicted last 3 states for GS and NS, respectively, vs. ground truth and Oracle using? base models. Each environment is shown by row. For CoDA, use the best Oracle result's CoDA version save space. See Appendix H for all the visualizations.
LEADS
CoDA-I
CoDA-I,
Data
Assignment
MSE
MAPE
MSE
MАPE
MSE
MAPE
All in One
4.1648.61 E-2
9.92#3,55
F.01:6.43.22
26.90=T0.65
4,1047.61 E:2
27,80&8.81
One per Env
2.28+1.81 E-3
$.110.500
1.72+0.53 E-3
27.6347
1.6640.82 82 B-3
25.51
Random
1.72+0.53 E-3
6.87=0.13
L14#0.61 E-3
27.56+6.36
1,0500.54
20.6546.31
Dynalnfer
5.77:1.46
2.8420.13
8.3720.94 E-5
10.16:0.04
8,4922007 E-5
10.30:0.08
Oracle
1.6742.26 15-3
3.1640.37
5.85&1.24 B-5
10.2440.06
5.1243.17 E-5
10.24:0.04
All in One
4.5911.18 E-4
121.254197.54
2.8500555 E:3
2658.2011651.77
2.7640.72 E:3
3355.201135.44
One per Fav
3.59+2.71 F-44
450.20+368, 36
1.0840822 E33
6247.344817.0
1,1940.97 F33
5948.574935.77
Random
5.7340.86 E-4
2.84+0.89
Dynalofer
1.00=0.32 E-4
378.73+182.12
2.41=0.91 E-4
220.54e65.60
2.13#0.41 E-4
207. 997.966446.49
Oracle
2.2140.93 E-4
434.73+432.38
2.6620.79 E-4
302 68 25
2. 2.1000.88 89 E-4
230.62&118
All in One
1,25+2.04 lb-2
67,17/13.33
1,2520.20 E-2
218.66127..26
1.2910.29 E222
214.44=17.08
One per Eav
2.7842.08 E-2
96,2344.54
2.0420.68 E-2
214.38&15.19
2.4340.48 E-2
209 199.68118..
E-2
DynaInfer
7.52=0.76 E:3
50,93+8.83
9.27&1881 E-3
101.38
9.71+2.10 E-3
101.04=16.27
Oracle
1.16#0.68 B-2
57.35+17.90
7 7,4600.22 E-3
154,86+41.00
7.32+0.81 81 E-3
100.04=24.93
Table 2: Adaptation Experiment Results the LV, GS, and NS environments. Dynalnfer consistently outperforms other non-Oracle methods across all datasets and narrows the performance gap with the Oracle more effectively compared to in-domain generalization.
4.3.3 Assignments Convergence
We illustrate the probability of environment assignments with DynaInfer training time in Figure Initially, our model may default to random assignments due to unoptimized neural networks. However, the assignments quickly converge the true labels. Notably, systems with simpler dynamics, like LV compared to NS, enable quicker learning of base generalization methods, resulting in faster convergence of environment assignments.
4.3.4 Performance across Varying Number of Assumed Environments
As the true number of environments ((E,) might be unknown, we examine Dynalnfer's performance with varying assumed environments M in Figure Our findings show that prior knowledge of the true M beneficial: performance peaks when the assumed M aligns with the true count. Additionally, our model demonstrates robustness to evr-msstmations of M. Due to its ability to account for bias, our model effectively identifies trajectories from the same environment and remains robust to an excess of inaccurately trained neural networks, even when the assumed M is too large. The above observations suggest that incrementally increasing the number environments until peak performance achieved can be straightforward way to identify the true M. Lastly, DynaInfer consistently outperforms other non-oracle approaches when M is underestimated, except for the One-per-Fin baseline (which requires M equal to the number of trajectories and is computationally infeasible).

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Fpwch 1:60
Spoch 1000
Fauth 1002
Ezemh 1636
Freen 1460
1.0
9
g
13
13
i y
17
17
17 1 39 2o
St
St
i
22
B
0.0
1250099
12400099
12306799
123006
1240699
1246679
1234008
129466
Epoch 400
Epoch 820
Epoch 1230
Esoch 1600
Epoch 200ch 2000
Epoch Epoch 2400
Eaech 2800
Eaoch 3230
Esech 3609
Epach 4000
17
17
17
17
57
23
0.2
8
0.0
1234
1234
1214
1234
Figure 4: Environment assignment probability over time, averaged over runs, with LEADS base model (on LV (top) and NS (bottom); see Appendix G for GS). The assignment converges to the true label faster than the designated training steps. similar trend observed with the CoDA model.
NS
2-002
1112
10005
4244-
2,202
24124
Eowe6
Enowd
Enved
Eov-20
Enve4
Cobbi
25163
SOF4
Tap
1,0022
Erii 11
L3123
TapE
L2002
Figure 5: Performance across assumed number of environments with Dynalnfer(log-scal y-axis). The peak performance aligns with the true number of environments (bold x-axis) with high probability, and remains stable thereafter.
Related Work
5.1 Domain Generalization and Adaptation
Domain generalization (DG) seeks train model domains so that generalizes effectively to any out-of-distribution (OOD) target domain. DG methods assume data heterogeneity and use additional environment labels to develop models that remain robust across unseen and shifted test data. Many DG strategies focus on domain alignment, aiming to minimize divergence among source domains to achieve Nomain-invarian
multiple distinct but related source
representations [2] 14 29
31
40
Other approaches enhance the diversity of training data by augmenting source domains [ Additionally, some methods leverage meta-learning and invariant risk minimization for
33
39]
regularization, further enhancing generalization (20
Domain adaptation (DA) methods enable model generalization to target domains with shifted data distributions and are primarily classified into three categories. Instance-based methods reweight or adjust training samples to reflect the test distribution [5]
6 Feature-based approaches align feature
distributions across training and test domains 38
36 Model-based strategies focus on developing
models that are either robust to domain shifts specifically tailored for the target domain [25 10

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5.2 Generalization for Dynamical Systems
Generalization in dynamical systems remains underexplored in literature. Among the limited studies, LEADS emerges as novel multi-task learning framework that effectively generalizes across the functional space of dynamical systems [4]. Alternatively, CoDA optimizes within the parameter space, enhancing model adaptability and efficiency while accommodating increased environmental variability without requiring multiple distinct network trainings for each setting [17]. In contrast, DyAd is context-aware meta-learning approach that adjusts the dynamics model by decoding time- invariant context from observed states [41]. Despite its novelty, Dy Ad relies on potentially impractical weak supervision based on physics-derived quantities and uses Adaptive Instance Normalization, which may degrade performance.
Currently, three notable weaknesses prevail in generalization works for dynamical systems. First, there is an assumption that prior knowledge about the target domain exists, and without it, most generalization methods would fail [TT]. Second, the predominant use of the mean squared error loss function is inadequate for evaluating the reconstruction accuracy of chaotic systems. Lastly, the influence of unlabelled trajectory data the process of learning generalizable dynamical systems remains both unexplored and unresolved - a gap this paper examines for the first time. While switched systems learning methods [19] infer modes by classifying individual data points (analogous to environment inference), our work operates on trajectories governed by ODEs or PDEs.
Conclusion
We propose an environment inference method that improves the understanding and generalization of complex dynamical systems across various environments without using manually labeled data. Dynalnfer infers environment labels directly from training data, overcoming the challenges associated with explicit annotation. Theoretical analysis ensures convergence of Dynalnfer, and experiments show Dynalnfer often surpasses non-oracle methods and matches or exceeds oracle performance.
Future research could unfold along the following promising directions: First, to improve generaliza- tion in chaotic systems, MSE-based methods could be replaced with more suitable metries such the sliced Wasserstein-1 distance, which would require developing a tailored inference model. Similarly, effectively inferring environments for dynamics with complex boundaries remains significant open problem, as current learning methods often oversimplify boundary conditions. Furthermore, for systems requiring interpretable parameters, our method could be extended to jointly optimize environ- ment labels and physical coefficients. To improve convergence, coordinate optimization techniques may help escape local optima for objectives that are convex in individual variable blocks [3] 30 Finally, techniques such as adaptive early stopping, dynamic batching, and membership functions could further enhance training efficiency.
Acknowledgments and Disclosure of Funding
This work was supported by NSFC (No. 62425206. 62141607, 62525213), Beijing Municipal Science and Technology Project (No. Z241 100004224009), and Big Data and Responsible Artificial Intelligence for National Governance, Renmin University of China. We also extend our thanks to Renzhe Xu and Hao Zou for their insightful discussions.
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A Proof of Proposition 3.1
Proof. Due the operation in Equation In addition, according to the operation in Equation 4 we must have
must have
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dl.
JEE
dt.
As result, since the regularization terms (i.e., the M( Rg (0fr), (())) remain the same, we must have
term) in R. Rezzi xb()) and
Rgcr+1 (11))(()) s Rain (010))(1)))
Now we have
Rair+ay (oirt1), (((+t)))((+t))) S Rato+iy (06) < Bgco (di), (b(7)
Now consider the second part of the proposition. Define the following space of 0
H A (0, =è E [M] such that Re(0, Φ) = min Re $0)}
0".
Note that by the assumption, we must have [H1] < co. Now define A as
dæt h (1):0,00 0z)
A
:t: [N],e [M], (0, Φ) e H n
dt
Note that since HH| < co, we have IA| < co. C' is defined as
C min |a - b).
ab
Since [A| < oo, we must have C Rg(etn (0(r+1), g(rtl)) (((t)) < Rgto (B(r), (())), have
0. addition, note that when r > 1 and * ((r) Otherwise we will
we must have è @(++1)
Rg(e1 (glr+1) (b(t1))))
= Rem (glo+1),
(00t,,,,())) ((()))
= Rgm (gir) (
As result, there exists E [N] such that a! ((++1) # E(r), By the choice of a! (r+1)) we must have
dx
s4 p())
(0())
||. dt
dt
e((r)
tET
#
dxp
h
rt: Q(r)
(o e(r+1))
dt.
dt
IET
As result, by the definition of C, we
have
Jest
dxf
(s(r))
) dl
dt
E((r)
dxt
Jter
dt
(1)
1(1
di C.
Therefore, we must have
Reo+t (glnt2), (10+t)),(++111) R2c0 (Bir) (6)))))>)))) -C
Now the claim follows.
21

---

B Real-world Experiment
We further evaluated our method using a real-world robot motion trajectory dataset [T6]. This dataset comprises three distinct motion patterns: (1) drawing "S" shapes, (2) placing cube shelf, and (3) drawing large "C" shapes. In our experimental framework, each of these patterns is treated as a distinct environment. The objective of this evaluation is to assess whether our method can accurately infer the underlying environment, thereby supporting the learning of generalizable neural network for simulating their dynamics. To ensure consistent input structure across environments, all trajectories were projected into a two-dimensional space. The dataset was partitioned into training and test sets with ratio of 6:1. Furthermore, each trajectory was temporally resampled to standardized length of 100 time steps for smoothing and uniformity following the practice in [44].
The empirical results are presented in Table In summary. these findings collectively suggest that our approach maintains its efficacy on real-world data, thereby substantiating its practical utility and robustness.
LEADS
CoDA-i
CollA-yy
Data
Assignment
MSE
MSE
MAPE
MSE
MSE
MAPE
MSE
MSE
MAPE
F:1
7.41
40.22-1.84
7.14
7.0+001
29,4843.15
7.17 F-2
74THUAR
39.26+22.33
One pez Env
4.15E-4
4.9113.50
6.6812441
8.68 E-4
9.11.0.11 E-4
5.6711.01
8.18 E-
8.1300399
5.7311.19
RM
Ranlom
7.20 E-2
7.38 E-2
50.01s1.05
7-12
7,39:0.01 E-2
48.87+1.81
7.09 E-2
7 7.5900.00 E-2
48.86+2.54
Dynalnfer
4.74
9.57
3.2712.36
2.02:1.66
Oracle
E-5
7.026076BB
1
1-78
3.190024 E-5
1,2610.06
1.99 E-5
72400.18 E-5
1,21:0.08
Table 3: In-domain Experiment Results on the Robot Motion.
C Sensitivity Analysis to |
To assess the robustness of our method to the number of underlying environments, we conducted sensitivity analysis using the LV system. We systematically varied the true number of environments, [Eol. from 2 to 16 and evaluated the performance of our model against the Oracle baseline that has privileged access to the true environment labels. The results are presented Table DynaInfer, achieves performance comparable to the Oracle across all tested values of |,, This empirical evidence demonstrates that our approach is robust to variations the quantity of environments.
Table 4: Test Mean Squared Error the LV system for varying [E
E
2
3
5
6
7
8
Dynalnfer Oracle
2.12E-5
4.22E-5
6.77E-5
6,24B-5
8.81E-5
6.70E-5
6.54E-5
2.76E-5
3.99E-5
5.75E-5
5.20E-5
7.24E-5
8.14E-5
5.74E-5
E,
10
11
12
13
14
15
16
DynaInfer
6.33E-5
8.34E-5
2.24E-4
1.34-4
9.69E-5
1.97E-4
3.43E-4
2.73E-4
Oracle
7.60k-5
9.31
1.85H4
1.23E-4
1.21H-4
1.71H-
2.561:4
2,48H-4
D Robustness over M
We assessed our robustness to the overestimation M, the assumed number of environments, in the LV system. The system's true value is M = 9. with each environment containing four trajectories. In our experiments, we varied the assumed M from to 36. The results, shown in Table|5] demonstrate that Dynalnfer effectively adapts to this overestimation. The learned label count consistently converged to a range 8- 11, which aligns closely with the ground-truth value.
Furthermore, we conducted comprehensive analysis by sweeping M across its full spectrum, from M = 1 to M #trajectory. As shown in Table the optimal performance was achieved at M and the model maintained robustness even when M exceeded this value.
22

---

Table 5: Learned Label Counts for different values of M averaged over runs
M
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
" Labels
5
6
7
8
9
8.8
9
10.2
9
9
10
10.8
10
11
11
10
22
25
26
27
28
29
30
31
32
33
34
35
36
Labels
10.8
11
9
10 10.2
11
11.2
9.2
10.8
11
Table 6:
Test MSE over the full spectrum of M for the LV system
M
MSE
M
MSE
M
MSE
M
MSE
7.41E-2
10
2.41E-4
19
2.60E-4
28
2.04E-4
2 4.01E-2
3.71E-4
20
1.36E-4
29
3.65E-4
3
2.47E-2
12
2.57E-4
21
1.21E-4
30
1,42E-3
4
1.4E-2
1.63E-4
22
2.45E-4
31
3.84E-4
5
7.02E-3
14
2.24E-4
23
9.06E-5
32
1.16E-4
6
4.24E-3
15
1.15E4
24
2.38E-4
33
3.89E-4
7
2.59E-3
1.76E-4
25
8,27E-5
3.10E-4
8
1.21E-3
17
1.83E4
26
1.06E-4
35
1.72E-4
9
7.93E-5
18
3.60E-4
27
3.71E-4
36
2.77E-4
E Environment Specification
We conducted experiments on server equipped with a 64-core CPU, 256 GB of RAM, and eight 24GB RTX-3090TI GPUs. The Dynalnfer framework was implemented using PyTorch [27) All NN params in our method are randomly initialized.
Lotka- VolterV (LV) 23 The system models the dynamics between prey-predator pair in an ecosystem, captured by the following ODE:
dm/dt = am 3mn, dn/dt ômn - y
where m, n represent the population density of the prey and predator, respectively, and a. 3, 6, are the interaction parameters between the two species. The system state is defined as 2 (m,. n,
R. with initial conditions (ma, nn)) sampled from a uniform distribution p(zn) = Unif 3 The environment is defined by dynamics parameters 0. (Ore/ Bey 7e/6.) θ sampled uniformly
from the set e. We simulate trajectories over a temporal grid with A/ = 0.5 and a horizon 7 10.
At test time, environment labels are inferred from the prediction bias observed over initial segment of the trajectory of length At.
The model uses simple difffsion equations to effectively study complex pattern formation in chemical and biological systems, following underlying PDE dynamics:
Gray-Scott (GS) 28
0m/0L D.. mnn F(1 (1-m),
Ôn/ôt D, 2 - k)mm
where m, n represent the concentrations of two chemical components in the spatial domain S with periodic boundary conditions, and Dm. D,, are their constant diffusion coefficients, and F. k are the reaction parameters that govern the spatio-temporal dynamic patterns. S is 2D space of dimension
32 x 32 with spatial resolution of As = 2. The system state x = (m,, mt) E 2x2322 We define the initial conditions (mo, no) p(2r)) by uniformly sampling three two-by-two squares, which activate the reactions, from S. (ma, no) = (1 - e, <, with e = 0.05 inside the squares and (mo, Rn) (0, 1) outside the squares. The environment is defined by dynamics parameters sampled uniformly from the environment distribution Q on θ. We simulate trajectories on temporal grid using timestep of At 40 over a horizon of T' 400. At test time, environment labels are inferred from an initial segment of length At.
(Fe, k.) E θ.
Navier-Stokes (NS) 22
The Navier-Stokes PDE describes the motion of viscous fluid substances:
0m/Ot= - vAm + Ve
23

---

where n. is the velocity field, m = V x is the vorticity, both m lie in spatial domain S with periodic boundary conditions, is the viscosity (fixed le 3) and is the constant forcing term in the domain S.
The system state is characterized by of = my E I2 as initialized per [22]. The enviromment is determined by uniformly sampled forcing term E (g.. We simulate trajectories across a temporal interval At = over a horizon inferred from an initial segment of length 2At.
10. At test time, environment labels are
The parameters for LV, GS and NS systems are respectively given in Table and 9
Table Parameters LV Systems
Params.
Train
Train 2
Train
Train 4
Train 5
Train
Train 7
Train
Train 9
Adapt 1
Adapt 2
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.7
0.6
0.5
0.75
0.5
0.5
0.75
0.75
0.8
0.7
0.5
0.5
0.5
0.5
0.5
0.5
0.75
0.75
0.5
Table 8: Parameters of GS Systems
Params.
Train
Train
Train
Adapt
Adapt
F
0.037
0.03
0.039
0.033
0.036
0.06
0.062
0.058
0.059
0.061
Table 9: Parameters of NS Systems
Train
0.1 (sin(2 +1) cos (X+
Train
0.1 * sin(2m(X++))) + cos * +X+2Y))
Train
0.1 (sin(m((X+Y)) + cos(2 (2X+
Train
0.1 (sin(2((X++2)) + cos(2n (2X+Y))
Adapt 1
0.1 (sin(2:(2X + +Y) cos(2n (X (X))
Adapt 2
0.1 (sin(2 +122x+)))) cos(2 (2X +X)+
F
Centroid-like NN Behavior
We conducted an additional experiment measuring the MSE loss for each training sample across different neural networks (NNs). The results, shown in Table 10 yielded two key findings: first, each training sample has uniquely best-fit NN where it attains minimal loss. Second, this optimal NN generalizes effectively to broader cluster of same environment. This alignment between "centroid- -ikee" NN and environment-speecfie trajectory clusters strongly reinforces our methodological rationale.
those originating from the
24

---

Traj.
NNI
NN2
NN3
NN4
NN5
NN6
NN7
NN8
NN9
Assign.
2.06c-01
1.18c-01
2.80c-01
7.76e-06
8.34c-02
1,59--01
3.14c-01
2.50c-01
2.35c-01
2.05c-01
1.21c-01
2.66c-01
9.35c-06
8.42c-02
1.53c-01
3.36c-01
2.40c-01
2.26c-01
2.05c-01
1.21c-01
2.65c-01
1.03c-05
8.41c-02
1.52c-01
3.35c-01
2.39c-01
2.25c-01
4
2.45e-01
1.51e-01
3.85c-01
1.98c-05
1.13e-01
1.89c-01
4.05e-01
3,10c-01
3.12e-01
5
3.31c-02
1.65c-01
1.12c-01
8.32c-02
5.36c-06
5.42c-02
3.68c-01
7.26c-02
1.37c-01
3.15c-02
1.87c-01
1.07c-01
8.46c-02
4.26c-06
5.57c-02
4.21c-01
6.80c-02
1.44c-01
3.14e-02
1.86c-01
1.06c-01
8.46c-02
4.50e-06
5.52c-02
4.19c-01
6.74e-02
1.43e-01
2.84e-02 8
2.68e-01
2.13e-01
1.14c-01
2.22e-05
9.57e-02
5.38c-01
1.18e-01
2.36e-01
7.60c-06 9
2.32c-01
5.48c-02
2.07c-01
3,36e-02
4,44e-02
4.22c-01
1.91e-02
1.18c-01
10
6.45c-06
2.64c-01
6.07c-02
2.05c-01
3.12c-02
5.176-02
4.93c-01
2.10c-02
1.37c-01
11
6,65c-06
2.63c-01
6.02c-02
2.05c-01
3.11c-02
5.14c-02
4.91c-01
2.07c-02
1.36c-01
12
1,95c-05
3.66c-01
1.61c-01
2.45c-01
2.920-02
9.83c-02
6.31c-01
7.05c-02
2.32c-01
13
2.31c-01
1.35e-05
1.56c-01
1.16c-01
1.64c-01
8.52c-02
5.28c-02
1.84c-01
7.0c--02
14 2.63e-01
1,39e-05
1.64e-01
1.22e-01
1.87e-01
9.48e-02
5.95e-02
2.02c-01
7.16e-02
15
01
14
1.63e-01
1.22c-01
1.86e-01
9,44c-02
5.93e-02
2.01c.01
0.13e-02
16 3.65e-01
1899-05
2.33e-01
1.50c-01
2.67c-01
1.19c-01
6.96c-02
2.56c-01
1.08c-01
17
4.22e-01
5.35e-02
2.43e-01
3.15e-01
3.68e-01
1.95e-01
2.78c-05
3.17e-01
1.13e-01
18
4.91e-01
5.91e-02
2.77c-01
3.35e-01
4,19c-01
2.27e-01
2.12c-05
3.66e-01
1,30c-01
19
4.89c-01
5.89c-02
2.76c-01
3.34c-01
4.176-01
2.26c-01
2.08c-05
3.64c-01
1,30c-01
20 6.28c-01
6.90c-02
3.26c-01
4.04c-01
5.35c-01
2.56c-01
1.99c-05
4.18c-01
1.52c-01
4.36c-02
8.60c-02
1.90c-02
1.58c-01
5.31c-02
7.09c-06
1.96c-01
1.98c-02
2.19c-02
22 0.
9.53e-02
1.74c-02
1.54c-01
5.58c-02
8.08c-06
2.28c-01
2.09c-02
2.34e-02
23
0.13e-02
9.49e-02
1.73e-02
1.53e-01
5.53e-02
7.86e-06
2.27c-01
2.08e-02
2.33e-02
24
9.81c-02
1.19c-01
3.84c-02
1.89c-01
9.61e-02
155e-05
2.56e-01
2.72c.02
3.4c.022
25
1.176-11
7.14c-02
2.46c-02
2.35c-01
1.36c-01
2.11e-02
1.15c-01
5.476-02
8.69c-06
26
1.37c-01
7.21c-02
2.82c-02
2.26c-01
1.44c-01
2.36c-02
1.31c-01
6.42c-02
5.20c-06
27
1.36c-01
7.18c-02
2.81c-02
2.26c-01
1.43c-01
2.35c-02
1.31c-01
6.39c-02
5.32c-06
28
2.32c-01
1.8e-011
3.35e-02
3.12c-01
2.37c-01
3.47c-02
1.52c-01
7.46e-02
8.12c-06
29
1.90e-02
1.84-001
1.02e-02
2.49c-01
7.19e-02
2.00e-02
3.17e-01
2.50e-06
5.57e-02
30
2.07-002
2.03c-01
1.15c-02
2.41c-01
6.79c-02
2.11c-02
3.68c-01
2.91c-06
6.44c-02
31
2.04c-02
2.02c-01
1.14e-02
2.40c-01
6.74e-02
2.102-02
3.66e-01
3.05e-06
6.41e-02
32
7.07e-02
2.55e-01
2.04e-02
3.09c-01
1.20c-01
2.74e-02
4.17e-01
8.74e-06
7.41e-02
33
5.48c-02
1.56e-01
1.22e-06
2.80c-01
1.11e-01
1.89c-02
2.44c-01
1.00c-02
2.53e-02
34
6.04e-02
1.65e-01
3.09c-06
2.67e-01
1.07c-01
1.76e-02
2.79c-01
1.14c-02
2.83e-02
35 5.99c-02
1.64c-01
3.10c-06
2.66c-01
1.06c-01
1.75c-02
2.78c-01
1.14c-02
2.82c-02
36
1.61e-01
2.32c-01
1.20c-05
3.85c-01
2.14c-01
3.90c-02
3.26c-01
2.06c-02
3.35e-02
Table 10: Training MSE for Trajectories by each Neural Network and Label Assignment
25

---

G
Environment Assignment Convergence on GS
Figure presents the temporal evolution environment assignment probabilities using the LEADS base model on GS.
Epoch Epoch 7000
0.8
13
11
02
Figure 6: Environment Assignment Probability over Time with LEADS as Base Model on GS. Despite initial inaccuracies due to complex dynamics, the assignment ultimately converges to the correct label.
H Plots on Learned Dynamics
We present the recovered test trajectories produced by the learned neural network. Figures 7 S and illustrate GS; Figures TO I1 de NS; and Figure 13 shows LV. A close examination reveals that the trajectories predicted by DynaInfer closely align with both the Oracle and the ground truth for the selected systems.
Figure Comparison final GS states predicted by DynaInfer (bottom) and Oracle (middle) against the ground truth (top) with base model LEADS.
26

---

Figure 8: Comparison final GS states predicted by DynaInfer (bottom) and Oracle (middle) against the ground truth (top) with base model CoDA-(1.
Figure Comparison final GS states predicted by DynaInfer (bottom) and Oracle (middle) against the ground truth (top) with base model CoDA-I2.
27

---

Figure 10: Comparison of final NS states predicted by Dynalnfer (bottom) and Oracle (middle) against the ground truth (top) with base model LEADS.
Figure 11: Comparison of final NS states predicted by Dynalnfer (bottom) and Oracle (middle) against the ground truth (top) with base model CoDA-I1-
28

---

Figure 12: Comparison of final NS states predicted by Dynalnfer (bottom) and Oracle (middle) against the ground truth (top) with base model CoDA-12-
29

---

1.25
1.2
1.2
1.0
1.1
Truth
1.0
Oradie
0.8
.75
Dynainfer
0.5
1.5
.25
1.0
1.4
2.0
1.2
1.4
1.2
1.5
Truch
0.8
Dracle
Dynolnfer
1.0
0.6
0.8
0.5
0.4
0.2
0.50
0.75
1.00
0.5
1.0
0.25
0.50
0.75
1.00
1.2
1.25
1.2
10
1.1
Tuth Oracle
0.8
0.75
1.0
Dynalnter
0.5
1.0
1.0
1.4
2.0
1.2
1.4
1.5
1.0
Truth
0.8
Oracle
Dynainfer
0.8
0.5
0.4
0.2
0.50
0.75
1.00
0.5
1.0
0.25
0.50
0.75
1.00
1.2
1.25
1.2
1.1
Truth
1.0
Orscle
0.8
3.75
Dynainter
0.5
1.0
1.0
1.4
2.0
1.2
1.4
1.2
1.5
1.0
Truch
0.0
Dracle
Dynainfer
0.8
0.5
0.4
0.2
0.50
0.75
1.00
0.5
1.0
0.25
0.50
0.75
1.00
Figure 13: Comparison of predicted LV trajectories from environments by DynaInfer (green) and Oracle (red) against the ground truth (blue) with base models LEADS (top), CoDA-lj (middle), and CoDA-l2 (bottom).
30


Reviews:
[{"review_id": "wxBW8E2SK6", "reviewer": "Reviewer_jxRD", "summary": "This paper presents an algorithm to learn dynamic systems for mixed sequence data without environment labels. The assumption of the algorithm is that dynamic models show similar loss patterns under the same environment configuration. Based on this, the author uses an iterative pipeline, similar as K-means, to update the environment setting (centroids) and the dynamic model.", "questions": "I have several more questions:\n1. Is the pipeline sensitive to more complex / different boundary conditions?\n2. In NS equation setting, is $\\xi$ a constant force or steady force? Or in other words, do you assume the function known for this setting, and you just cluster trajectoies to different function settings?\n3. What is the $\\Omega$ term in your paper? This part is missing so I am kinda confused.", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths\n1. This paper is working on an important topic for dynamic system learning\n2. The proposed framework is simple and intuitive, which is flexible to many different dynamic systems. \n3. The model shows promising performance on selected dataset.\n\nDespite the strengths above, the paper also has the following weaknesses:\n1. Although the pipeline does not require the real annotation of environment, a finite environment set is required. For those continuous environment settings, the pipeline requires K exact proposals, which is a too-strong assumption in real world. The experiments shown in the paper also follow this requirements. The tolerance on the error of environment proposals is not assesed. \n2. All the experiments are taken with synthetic dataset with no real-world dataset is tested. \n\nCurrently I think this is a paper with a tidy idea, but the potential ability for broader application is limited due to the too-strong assumption. Since I’m not primarily working on this field, I am not sure whether my concern is really important to this community. So I’m open to change my recommendation after reading other reviewers’ opinion and the rebuttal from the authors.", "was_revised": true, "final_justification": "The author has addressed all my concerns. I believe this paper includes an instructive idea and a potential pipeline to address environment when learning dynamical system. \n\nAt the very beginning, I misunderstood the paper requiring including gt as candidates environment. This point is clarified successfully by the authors. So I am willing to increase my rating from 3 to 5. The relatively simple experiment in real-world dataset prevent me from giving rating 6 recommendation.", "rebuttal": "We sincerely appreciate the reviewer’s time and valuable feedback on our work. Below, we provide a point-by-point response to their comments. For clarity, some raised questions have been condensed to focus on key points. In this rebuttal, new references are labeled as [N1], [N2], etc., while existing references in the manuscript retain their original numbering.\n\n>  **[W1] Although the pipeline does not require the real annotation of environment, a finite environment set is required [...] which is a too-strong assumption in real world. [...]. The tolerance on the error of environment proposals is not assessed.** \n\n**Response:**\nWe sincerely appreciate the reviewer's valuable feedback regarding environmental assumptions in our framework. We have addressed these concerns through **additional experiments and analysis, now included in the Appendix**. Below we provide a detailed response:\n\n#### 1. **On the finite environment assumption:**  \n- Our work addresses **domain generalization for dynamic system learning**, where existing literature typically uses a modest number of discrete environments (usually $|\\mathcal{E}_o| \\leq 10$) [14][37][N1][N2]. This is a practical constraint, as real-world data collection or simulation across numerous environments is often challenging. \n- For example, in robot motion learning with varying objectives (e.g., grasping, transportation) or conditions (e.g., payload weights, surface friction coefficients), acquiring exhaustive environment variations is difficult. \n- Additionally, an excessively large $|\\mathcal{E}_o|$ could introduce additional generalization challenges in neural network training, apart from environment assignment. We leave this broader discussion to future work, as it falls outside the scope of this paper.\n\n#### 2. **Environment Proposal Error Analysis:**  \n- We have carefully conducted additional experiments on the LV environment, **evaluating performance with true environment counts $|\\mathcal{E}_o|$ ranging from 2 to 16**. The results (shown in test MSE, as below) compare our model against the Oracle baseline. Our performance **matches Oracle for all counts**, and the learned label converges to real ones.\n\n|M|2|3|4|5|6|7|8|\n|---|---|---|---|---|---|---|---|\n|DynaInfer|2.12E-5|4.22E-5|6.77E-5|6.24E-5|8.81E-5|6.70E-5|6.54E-5|\n|Oracle|2.76E-5|3.99E-5|5.75E-5|5.20E-5|7.24E-5|8.14E-5|5.74E-5|\n\n|M|9|10|11|12|13|14|15|16|\n|---|---|---|---|---|---|---|---|---|\n|DynaInfer|6.33E-5|8.34E-5|2.24E-4|1.34E-4|9.69E-5|1.97E-4|3.43E-4|2.73E-4|\n|Oracle|7.60E-5|9.31E-5|1.85E-4|1.23E-4|1.21E-4|1.71E-4|2.56E-4|2.48E-4|\n\nThese experiments demonstrate our robustness to variations in environment quantity. Such table **has been added** in the Appendix.\n\n>  **[W2] All the experiments are taken with synthetic dataset with no real-world dataset is tested.**\n\n**Response:** We sincerely appreciate the reviewer's valuable advice regarding real-world dataset. We now justify the synthetic data usage and *provide additional experiments and analysis on real-world data, to be added in revision.\n\n#### 1. **Justification for Synthetic Data Usage:**\n- Our work tackles a **novel challenge** lacking direct benchmarks in existing open datasets. While prior domain generalization efforts in dynamic systems **often use synthetically combined data** to simulate real-world conditions, they rarely use real-world data that capture organic, unstructured complexities. Thus, synthesizing multi-environment datasets from diverse sources—as done in our work—is a well-established practice. \n\n#### 2. **Extra Validation on Real-World Data:**\n- Despite the scarcity of suitable real-world datasets for our problem setting, we have **added an extra  real-world robot motion trajectory dataset** [N3]. This dataset includes three distinct motion patterns:   (1) Drawing \"S\" shapes, (2) Placing a cube on a shelf; (3) Drawing out large “C” shapes. \n   - These three kinds of motions are represented as different environments. \n   - We treat these motions as different environments and evaluate whether our method can infer the environment to support the learning of a generalizable neural network capable of simulating their dynamics. \n   - Empirically, our experiments yield the following results:\n\n|Method|TrainMSE (LEADS)|TestMSE (LEADS)|TestMAPE (LEADS)|TrainMSE (CODA1)|TestMSE (CODA1)|TestMAPE (CODA1)|TrainMSE (CODA2)|TestMSE (CODA2)|TestMAPE (CODA2)|\n|---|---|---|---|---|---|---|---|---|---|\n|All in One|1.43E-2|1.56±0.35E-2|47.49±6.11|1.18E-2|1.29±0.17E-2|37.50±5.88|1.27E-2|1.43±0.11E-2|44.27±3.46|\n|OneperEnv|1.15E-2|1.39±0.03E-2|35.79±5.34|1.03E-2|1.07±0.10E-2|34.14±4.26|1.30E-2|1.37±0.62E-2|40.43±5.49|\n|Random|1.62E-2|1.73±0.11E-2|48.91±4.16|1.76E-2|1.85±0.13E-2|56.32±6.22|1.47E-2|2.11±0.21E-2|49.32±7.33|\n|DynaInfer|5.57E-3|7.32±2.12E-3|29.92±3.99|3.97E-3|4.72±0.87E-3|24.61±2.93|3.98E-3|7.10±2.23E-3|28.29±2.57|\n|Oracle|3.32E-3|5.18±2.27E-3|25.21±4.07|2.46E-3|3.53±0.54E-3|19.54±2.19|3.03E-3|6.17±2.28E-3|26.63±4.09|\n\n- In summary, these findings confirm that our approach remains effective on real-world data, reinforcing its practical applicability. We **have included these results** in the revised manuscript.\n\n>  **[Q1] Is the pipeline sensitive to more complex / different boundary conditions?**\n\n**Response:** We appreciate this insightful question. We interpret it in two possible ways, addressing each below:  \n\n#### **1. Boundary Conditions in Physics Systems**  \nIf the question concerns whether our method is sensitive to the *physical boundary conditions* of the dynamical systems studied:  \n- Our experiments **leverage periodic boundary conditions** in Navier-Stokes and Gray-Scott systems. While *mathematically simpler to implement*, they still exhibit **inherently complex dynamics** for base dynamic system learning methods. \n- While some CFD benchmarks feature even more complex boundary conditions, these dataset remain underexplored in current works on *domain generalization for dynamic systems*.  \n- We fully agree that investigating such cases would strengthen our framework’s impact. Since our method operates on trajectory-level loss patterns, it is theoretically applicable to arbitrary boundary conditions. We believe adapting it for complex boundary conditions is an exciting direction for future work. \n  - We **have added following discussion** in [Conclusion, Line 317]: \"Moreover, since existing dynamical system learning methods largely rely on periodic boundary conditions for efficiency and effectiveness, exploring robust environment inference approaches for complex boundaries presents a significant opportunity.\"\n\n#### **2. Heterogeneity Across Trajectories (Data Diversity)**  \nIf the question refers to *heterogeneity in trajectory distributions* across environments:  \n- **Moderate Heterogeneity:** Our main results demonstrate robustness to reasonable variations in trajectory hyperparameters (as in current setting).  \n- **Low Heterogeneity:** To further validate robustness, we conducted additional experiments (*to be included in revision*) with subtle environment-specific differences. Specifically, we tightened the parameter gaps between environments in the LV system from the original 0.25 (e.g., for Env 1 and 2, parameters was $(\\alpha, \\beta, \\delta, \\gamma) = (0.5, 0.5, 0.5, 0.5)$ and $(0.5, 0.75, 0.5, 0.5)$ to gaps ≤ 0.1 and 0.05. Results (averaged over 5 runs) are as follows:  \n\n|Method|Δ≤0.1|Δ≤0.05|\n|-|-|-|\n|DynaInfer|6.95E-5|3.94E-5|\n|Oracle|5.51E-5 |3.10E-5|\n\nThere are two key observations:  \n  - Our model still learns generalizable dynamics, matching oracle performance.  \n  - **Environment labels converge to ground truth** even in low-heterogeneity scenarios, thanks to our risk-minimization mechanism for latent environment inference.  \n\nBoth interpretations highlight valuable limitations and extensions. We thank the reviewer for prompting this discussion and have clarified these points in the paper.  \n\n>  **[Q2] In NS equation setting, is ξ a constant force or steady force? Or in other words, do you assume the function known for this setting, and you just cluster trajectories to different function settings?**\n\n**Response:** \n- To clarify, in all dynamic system considered in this paper, all the environment-specific system parameters, (e.g., the constant force $\\xi$ for the NS system), **are unknown to the model**. \n- If system parameters are known, environment labels could be trivially inferred by classifying trajectory parameters defining the governing dynamics. \n- We follow the realistic scenario in dynamic system learning where the environment-defining parameters are latent and must be inferred solely from observed trajectories.  \n\n>  **[Q3] What is the Ω term in your paper? This part is missing so I am kinda confused.**\n\n**Response:** We sincerely appreciate this question. \n\n- $\\Omega(\\phi_{e})$ is a regularization term applied to the environment-specific parameters $\\phi_{e}$ to promotes better generalization. It prevents trivial solutions where all meaningful dynamics are captured by $\\phi_{e}$ instead of the shared parameters $\\theta$.\n- The key ingredient for multi-environment learning is to decompose the neural network as $f_\\theta+g_{\\phi_{e}}$, and the shared component $f_\\theta$ should capture maximal shared dynamics across environments, whereas the unique parts $g_{\\phi_{e}}$ should only capture truly environment-specific characteristics. \n- Without $\\Omega$, optimization often converges to degenerate solutions where $\\phi_{e}$ absorbs all dynamics (making $\\theta$ meaningless).\n\nFinally, we sincerely welcome further discussion and appreciate any additional suggestions. \n\n[N1] Continuous PDE Dynamics Forecasting with Implicit Neural Representations. ICLR. 2023. [N2] Stochastic neural simulator for generalizing dynamical systems across environments. AAAI. 2024. [N3] Learning stable nonlinear dynamical systems with gaussian mixture models. IEEE Transactions on Robotics"}, {"review_id": "IOs9ZzBMjU", "reviewer": "Reviewer_ghmr", "summary": "The paper studies the problem of learning dynamical system models wherein the model is adapted to data coming from different environments. This is parameterized as a leraning problem wherein there are common parameters that are environment-independent and some environment dependent parameters. However, the environment labels for trajectories are latent and need to be inferred. The paper uses a k-means (or also EM-style) approach to solve this problem by alternating between learning an environment assignment and at the same time using the current environment assignment to learn new parameters. This is very much in line with k-means clustering. Some ideas from clustering are used to improve the learning scheme. It is demonstrated on a series of interesting benchmark examples.", "questions": "I would welcome your observations on some of the issues raised in my review.", "limitations": "I did not see a meaningful discussion but the paper is written in a careful and modest style that makes limitations clear as one reads.", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "The ideas in the paper are very interesting: I really enjoyed reading about the problem formulation and its translation into an alternating minimization framework. I found the writing to be crystal clear and the ideas to be very easy to comprehend.  The results are also very interesting.\n\nPerhaps, my biggest critique is that the paper ignores a lot of research into identical ideas from the world of switched systems regression.  Perhaps the connection is not well known, but the problem of learning switched systems from data is very similar if not identical in many respects. Lauer and his coworkers have been working on a very similar idea that they call k-linear regression -- it is principally meant for linear systems or learning non linear systems once the basis functions are identified. However, the ideas are very similar. I would recommend looking into Lauer's papers. These ideas are somewhat mainstream in the area of switched systems identification.\n\n@article{lauer:hal-00743954,\n  TITLE = {{Estimating the probability of success of a simple algorithm for switched linear regression}},\n  AUTHOR = {Lauer, Fabien},\n  URL = {https://hal.science/hal-00743954},\n  JOURNAL = {{Nonlinear Analysis: Hybrid Systems}},\n  PUBLISHER = {{Elsevier}},\n  VOLUME = {8},\n  PAGES = {31-47},\n  YEAR = {2013},\n  DOI = {10.1016/j.nahs.2012.10.001},\n  KEYWORDS = {switched regression ; system identification ; switched linear systems ; piecewise affine systems ; sample size ; nonconvex optimization ; global optimality},\n  PDF = {https://hal.science/hal-00743954v1/file/LauerNAHS12.pdf},\n}\n\nYou can find more details in the Springer book of Lauer and Bloch:\n\nHybrid System Identification: Theory and Algorithms for Learning Switching Models -- Lauer and Bloch, 2019.\n\n@article{BIANCHI2022110589,\ntitle = {A constrained clustering approach to bounded-error identification of switched and piecewise affine systems},\njournal = {Automatica},\nvolume = {146},\npages = {110589},\nyear = {2022},\nissn = {0005-1098},\ndoi = {https://doi.org/10.1016/j.automatica.2022.110589},\n}\n\n\nAlthough I agree that the setting of this paper is different from that of switched system (the label assignment is stationary in time in this paper whereas in switched system regression, it varies over time), I am just interested in understanding how the overall ideas connect -- the connections between the two settings are stronger than the differences.\n\n- Also, the optimization approach used here is well known to be problematic. A lot of studies have shown that for k-means regression. But specifically this sort of alternating minimization has issues. It can be shown to seldom converge to a KKT point and often gets stuck at saddle points. See this paper by Helton and Merino for an example of such a result -- \n Helton, J., Merino, O.: Coordinate optimization for bi-convex matrix inequalities. In: IEEE Conf. on Decision & Control(CDC). p. 36093613 (1997)", "was_revised": true, "final_justification": "Given the discussions surrounding the paper, I raised my score to accept but also reduced the confidence of my review. Clearly, I did not understand some of the original aspects of this paper.", "rebuttal": "We sincerely appreciate the reviewer’s time and valuable feedback on our work. Below, we provide a point-by-point response to their comments. For clarity, some raised questions have been condensed to focus on key points.  \n\n>  **[W1] Perhaps, my biggest critique is that the paper ignores a lot of research into identical ideas from the world of switched systems regression. Perhaps the connection is not well known, but the problem of learning switched systems from data is very similar if not identical in many respects. Lauer and his coworkers have been working on a very similar idea that they call k-linear regression -- it is principally meant for linear systems or learning non linear systems once the basis functions are identified. However, the ideas are very similar. I would recommend looking into Lauer's papers. These ideas are somewhat mainstream in the area of switched systems identification.** \n\n**Response:**\nWe sincerely appreciate the reviewer’s insightful feedback and the valuable references to the switched systems literature, particularly the work by Lauer. We have carefully reviewed these works and provide below a detailed discussion of their connections to and distinctions from our approach.  \n\n#### **1. Key Distinctions Between Problem Settings**  \n- **Switched System Learning (SSL)** primarily addresses:\n  - Identification of systems with discrete transitions between predefined subsystems.\n  - The switching mechanism is either known or learned as part of the modeling.\n  - Modeling of relatively simple subsystem dynamics (typically linear or affine).\n  - The primary objective is jointly estimating subsystem parameters and switching logic. \n\n- **Our Dynamic System Learning Generalization Work** differs fundamentally in that:\n  - **Absence of Switching** Our framework learns vector field functions (ODE or PDE) for continuous dynamical systems without switching; while SSL does;\n  - **Out-of-distribution and In-distribution**: Our primary focus is out-of-distribution generalization rather than in-distribution fitting, while SSL does;\n       -  Therefore, our optimization objective incorporate additional regularization terms specifically designed for domain generalization.\n  - **Non-linear Dynamics**: Our approach handles arbitrary nonlinear dynamics with complex boundary conditions (e.g., Navier-Stokes systems as in our work), while SSL typically handles switched linear or affine systems;\n\n#### **2. Technical Comparisons**  \nWhile both lines of approaches employ clustering-like mechanisms, there are crucial differences: \n\n- **Different Clustering**:\n     -  Switched systems methods (e.g., Lauer, Bianchi) cluster individual data points to identify subsystem boundaries. \n     - Our method clusters entire trajectories to learn the shared part that governs generalizable dynamics.\n- **Different Objectives**:\n     - The optimization objectives are fundamentally different (fitting accuracy vs.  domain generalization capability)\n\n#### **3. Potential Connections and Future Directions**  \n\nWe acknowledge several insightful connections in the suggested papers:  \n- The *k*-LinReg in Lauer's work also employs a min-min scheme, transforming the original least-squares switched linear regression problem into a minimum-of-error formulation. While our objective differs (leading to distinct theoretical guarantees), the overall methodology is analogous. Additionally, we could explore probabilistic evaluations of success rates for global optimality, inspired by their work, as future research.  \n- The two-phase constrained clustering approach by Bianchi introduces preference-space clustering for switched systems, which could inspire improved initialization for our environment discovery scheme. However, since their method computes preference vectors over individual data points, direct application to our setting—where we handle trajectories governed by ODEs/PDEs—is infeasible. Nevertheless, adapting their approach by considering the likelihood in temporal differences among trajectories could be promising, and we will explore this direction in future work.  \n\n#### **4. Additions to Manuscript** \nConsidering the insightful connections in switched systems, we would:\n- Discuss these connections and distinctions in the related work section. We **have added** following discussion in Line 306:\n   - \"While existing methods for learning switched systems, like *k*-LinReg [38], infer modes by classifying individual data points (analogous to environment inference), our work operates over trajectories driven by ODEs or PDEs. Despite this distinction, their methodology may still provide valuable insights for future developments in our framework.\"\n- Explore the suggested initialization method in future work. We **have added** following discussion in Line 320:\n   - \"Certain membership functions, such as the preference vector approach [39], could also facilitate environment label convergence.\"\n\n>  **[W2] Also, the optimization approach used here is well known to be problematic. A lot of studies have shown that for k-means regression. But specifically this sort of alternating minimization has issues. It can be shown to seldom converge to a KKT point and often gets stuck at saddle points.** \n\n**Response:**\nWe sincerely appreciate the reviewer’s insightful feedback regarding potential optimization challenges. We acknowledge the concerns about convergence behavior in alternating minimization and would like to address them as follows:\n\n- **Empirical Validation**: Experiments show that environment labels consistently converge to their true values (see Figure 3 in the main text), with optimization loss behavior matching the oracle baseline (Table 1).  This observation also aligns with prior work in machine learning (e.g., HRM and KerHRM [N1][N2]), where similar alternating optimization between environment partitioning and model learning has proven empirically effective.\n- **Potential Improvement**: While our current implementation uses hard label assignments, we have previously experimented with soft-weight alternatives that demonstrated comparable performance. Building on this, we could further enhance the probabilistic approach by introducing perturbations to the soft weights, which may help escape local optima. We plan to investigate this direction in future work.  \n- **Discussion of Suggested Work**: We thank the reviewer for mentioning coordinate optimization, a technique used for problems where the objective is convex with respect to each block of variables but not necessarily jointly convex. While our setting differs slightly, we will discuss its applicability and include it as a future direction in the revised manuscript.  \n  - In Line 316, we have added following discussion: \"Regarding convergence, coordinate optimization - a technique particularly useful for objectives that are convex with respect to individual variable blocks - may help escape local optima.\"\n\nWe sincerely appreciate the reviewer’s overall positive assessment of our work and their constructive feedback. This is of great significance for further strengthening this article and the subsequent work.  We welcome further discussion on these points and would be grateful for any additional suggestions to improve our paper.\n\n[N1] Liu, J. et. al., 2021, July. Heterogeneous risk minimization. In International Conference on Machine Learning. PMLR.\n[N2] Liu, J. et. al., 2021, December. Kernelized heterogeneous risk minimization. In Proceedings of the 35th International Conference on Neural Information Processing Systems"}, {"review_id": "U4Amsls4RW", "reviewer": "Reviewer_dkAK", "summary": "This work presents DynaInfer, a bi-level algorithm for learning environment labels and generalized environmental dynamics at the same time. This work is motivated by the fact that the collected trajectories generated from various environments lack true labels, and thus, it is hardly possible to use them for training generalizable surrogate models modulated on such labels. Instead of using such ground truth labels, this work suggests to optimize such labels altogether during optimization. This framework is bi-level, because it optimizes the label assignments to the trajectories in the first stage, and then optimizes the surrogate model parameters in the second stage based on the label assignments. This process is analogous to the K-means clustering, as the authors explain. The authors provide theoretical proof about the monotonic decrease of the approximation error using this bi-level approach. Then, the authors provide experimental results on three dynamical systems, which show DynaInfer’s efficacy over the baseline label assignment approaches.", "questions": "In my understanding, the number “M” is a hyperparameter that interpolates between “All-in-one” (M=1) and “One-per-Env” (M=# trajectory) strategy. If it is, I think it would be also interesting to see how the performance interpolates between them based on M. Then, I think it becomes easier to claim that DynaInfer is superior than both of these strategies as they are lower bounds of DynaInfer’s performance.", "limitations": "As I pointed out, the computational cost and the presence of hyperparameter M seems to be a potential limitations of this work.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n\n- The authors provided a theoretical analysis of their algorithm, and the experimental results look good. DynaInfer consistently outperforms the other baseline label assignment strategies, and the figure about convergence looks neat.\n\nWeaknesses\n\n- I feel some clarifications are needed about methodology. Specifically, in section 3.1, how do we solve the minimization problem posed in Eq. 4? Do we just try every label and select the best one? If that is the case, I believe there should be a computational cost analysis, because such brute force approach would incur more overhead. I think then the computational cost would increase linearly to M, which could be problematic if we use DynaInfer for a large dataset.\n- Even though the authors mentioned that DynaInfer consistently outperforms the other baseline label assignment strategies even when they use underestimated “M”, it seems not to be true. For instance, when M = 3, LEADS performance for LV is 2.47E-2, which is worse than the “One-per-Env” strategy (4.15E-4). I’m a little bit concerned about this necessity for prior knowledge of “M”. To truly make this approach free from prior knowledge, it would be good if there is one more strategy to optimize “M” automatically (even though the authors suggested a manual strategy for it).\n- It is not a main weakness, but it would be better to highlight the difference between the results in Figure 2 to show how DynaInfer outperforms the other methods.", "was_revised": true, "final_justification": "This paper solves the problem when there is no ground truth labels for the trajectories, which seems to be a fairly new problem in this field. Considering such contribution and as the authors provided detailed responses to my concerns, I will raise my score to accept.", "rebuttal": "We sincerely appreciate the reviewer’s time and insightful feedback on our work. Below, we address the raised questions and concerns point by point. For clarity, some responses have been condensed to highlight key points. We welcome further discussion and appreciate any additional suggestions.\n\n>  **[W1] I feel some clarifications are needed about methodology. Specifically, in section 3.1, how do we solve the minimization problem posed in Eq. 4? Do we just try every label and select the best one? If that is the case, I believe there should be a computational cost analysis, because such brute force approach would incur more overhead. I think then the computational cost would increase linearly to M, which could be problematic if we use DynaInfer for a large dataset.** \n\n**Response:**\nWe thank the reviewer for raising this crucial implementation concern. Below we clarify DynaInfer’s computational efficiency and justify its design trade-offs:\n\n#### 1. **Optimization Strategy** \n- While the reviewer correctly notes that inferring environment labels involves evaluating trajectories across multiple networks, this process is fully parallelized across both environments and batches, avoiding sequential overhead. \n- DynaInfer’s alternating optimization comprises two stages: \n  - Inference Stage: Complexity scales with assumed number of environments (M), and batch size (N), but crucially, it avoids the far greater cost of computing the full training graph (as required in optimization stage).\n  - Optimization Stage: Complexity scales linearly with (N).\n\n#### 2. **Empirical Runtime Analysis**  \nWe have included a wall-clock time comparison (seconds) on the LV system using LEADS as the base model, evaluating DynaInfer against the Oracle baseline (which requires no inference). The DynaInfer results are shown for the standard setting (*M*=9) and an exaggerated scenario (*M*=36) to stress-test efficiency:  \n\n| Setting       | Oracle | M=9   | M=36  |  \n|--------------|--------|-------|-------|\n| Time (s)     | 2526| 3258 | 4849 |  \n| Overhead     | —      | +28.9% vs. Oracle | +48.9% vs. M=9 |  \n\n- **Justifiable Overhead**: DynaInfer incurs only a **28.9% runtime overhead** (3258s vs. Oracle’s 2526s) under the real-world *M*=9 setting. This modest cost is outweighed by DynaInfer’s ability to **match Oracle performance *without* labeled data** (Tables 1–2, main text), eliminating manual labeling entirely.  \n- **Sub-linear Scaling**: Even under the exaggerated *M*=36 scenario (4× the true environment count, and not practical in real world), runtime increases by only **48.9%** versus *M*=9. \n\n#### 3. **Future Strategies**  \n- We agree with the reviewer that runtime is worth optimizing. In the revised manuscript, we have expanded discussion on efficiency:\n   - *Adaptive Early Stopping*: Halting inference once label assignments stabilize.\n   - *Dynamic Batching*: Prioritizing uncertain trajectories for re-evaluation.\n- These directions could reduce overhead while retaining DynaInfer’s benefits. We thank the reviewer for highlighting this nuance and will ensure above discussions are explicitly included in the final draft.\n\n>  **[W2] Even though the authors mentioned that DynaInfer consistently outperforms the other baseline label assignment strategies even when they use underestimated “M”, it seems not to be true. [...]. I’m a little bit concerned about this necessity for prior knowledge of “M”. To truly make this approach free from prior knowledge, it would be good if there is one more strategy to optimize “M” automatically (even though the authors suggested a manual strategy for it).** \n\n**Response:**\nWe sincerely appreciate the reviewer’s insightful critique regarding the need for prior knowledge of M. We address these concerns below and propose concrete improvements:\n\n#### 1. **Correction and Clarification** \n-  We acknowledge the reviewer’s observation regarding the performance of DynaInfer with underestimated *M*. Our original claim inadvertently omitted the edge case of the *One-per-Env* baseline, which requires *M* to match the number of trajectories—a scenario that incurs prohibitive computational costs and contradicts the goal of domain generalization in dynamic systems. We apologize for this oversight.  \n- The corrected statement in the manuscript now reads: “Lastly, DynaInfer consistently outperforms other non-oracle approaches when M is underestimated, except for the One-per-Env baseline (which requires M equal to the number of trajectories and is computationally infeasible).” \n\n#### 2. **Possible Automatic M Identification Strategy**  \nTo reduce reliance on prior knowledge of *M*, we’ve identified a practical and automatic strategy:  \n- **Gradual Increment with Early Validation:** We observe that an appropriate M yields measurable improvements even in early training stages (e.g., within the first 500 steps). Thus, M can be incrementally increased while monitoring validation performance, stopping when improvements plateau. This approach balances efficiency with robustness, avoiding the need for exhaustive manual tuning.  \n\n#### 3. **Empirical Support for Robustness over M**  \nWe conducted additional experiments on the LV system to evaluate robustness to *M* by setting *M* from 5 to 36 (which is the number of trajectories). \n\n| M        | 5    | 6    | 7  | 8    | 9    | 10  | 11 | 12   | 13   | 14   | 15 | 16   | 17   | 18  | 19 | 20   |\n|----------|------|------|----|------|------|-----|----|------| ------|------|----|------|------|-----|----|------|\n| **# Labels** | 5    | 6    | 7  | 8    | 9    | 8.8 | 9  | 10.2 | 9    | 9    | 10 | 10.8 | 10   | 11  | 11 | 10   |\n\n| M        | 21   | 22   | 23 | 24   | 25   | 26  | 27 | 28   | 29   | 30   | 31 | 32   | 33   | 34  | 35 | 36   |\n|----------|------|------|----|------|------|-----|----|------|------|------|----|------|------|-----|----|------|\n| **# Labels** | 10.8 | 11   | 9  | 10   | 10.2 | 11  | 10 | 10   | 10   | 11.2 | 10 | 9.2  | 10   | 10.8  | 10 | 11   |\n\n- As shown below (results averaged over 5 runs), DynaInfer adapts to the overestimation of *M*: The learned label count converges to a reasonable range for *M* (8–11), closely aligning with the true condition (true *M* = 9 in the LV system).\n- This also support the viability of automatic M selection. \n\nWe **have added such table and discussion** in revision.\n\n>  **[W3] It is not a main weakness, but it would be better to highlight the difference between the results in Figure 2 to show how DynaInfer outperforms the other methods.** \n\n**Response:**\nWe sincerely appreciate the reviewer’s constructive suggestion to more clearly demonstrate DynaInfer’s advantages over baseline methods.  \n\n- In our original submission, Figure 2 primarily focused on comparing DynaInfer’s performance against the *Oracle* baseline to emphasize its near-optimal capability, while omitting other methods for visual clarity (as their results were substantially weaker). \n- However, we fully agree that explicitly quantifying this performance gap would strengthen the paper’s impact.  Accordingly, we **have included a detailed visualization** in the revised Appendix to clearly showcase the differences.\n\n>  **[Q1] In my understanding, the number “M” is a hyperparameter that interpolates between “All-in-one” (M=1) and “One-per-Env” (M=# trajectory) strategy. If it is, I think it would be also interesting to see how the performance interpolates between them based on M. Then, I think it becomes easier to claim that DynaInfer is superior than both of these strategies as they are lower bounds of DynaInfer’s performance.** \n\n**Response:**\nWe sincerely appreciate the reviewer's insightful suggestion regarding the interpolation behavior of M. We have conducted extensive experiments across the full spectrum of M values (from 1 to the number of trajectories) on the LV system, with results that strongly support our claims.:\n\n|M|1|2|3|4|5|6|7|8|9|\n|---|---|---|---|---|---|---|---|---|---|\n|MSE|7.40E-2|4.01E-2|2.48E-2|1.04E-2|7.13E-3|4.24E-3|2.63E-3|8.25E-4|5.99E-5|\n\n|M|10|11|12|13|14|15|16|17|18|\n|---|---|---|---|---|---|---|---|---|---|\n|MSE|2.45E-4|2.33E-4|2.07E-4|1.63E-4|2.24E-4|1.15E-4|1.76E-4|1.83E-4|3.60E-4|\n\n|M|19|20|21|22|23|24|25|26|27|\n|---|---|---|---|---|---|---|---|---|---|\n|MSE|2.60E-4|1.36E-4|1.21E-4|2.45E-4|9.06E-5|2.38E-4|8.27E-5|1.06E-4|3.71E-4|\n\n|M|28|29|30|31|32|33|34|35|36|\n|---|---|---|---|---|---|---|---|---|---|\n|MSE|2.04E-4|3.65E-4|1.42E-3|3.84E-4|1.16E-4|3.89E-4|3.10E-4|1.72E-4|2.77E-4|\n\n- As shown in the tables below, MSE decreases sharply from M=1 (All-in-One) to M=9 (true number of environments), then stabilizes. The optimal performance occurs at M=9 and performance remains robust even with M overshoot. \n- The performance on the LV for different M, interpolates between M=1 and M=#trajectory. Similar trends were observed in the GS dataset (to be included in the appendix).\n- We would claim that DynaInfer is superior than both of these strategies as they are lower bounds of DynaInfer’s performance. We will add a new figure showing the performance-M curve for all datasets in the revised manuscript.\n\nFinally, we sincerely welcome further discussion and appreciate any additional suggestions."}, {"review_id": "CzQNKzyQ9c", "reviewer": "Reviewer_sBk5", "summary": "The authors introduce DynaInfer, a method to infer environment variables for dynamical systems directly from training data. This method allows to use multi-environment data and predict their dynamic response more accurately by inferring the environment variable. Experiments show improvments over other basic environment inference strategies.", "questions": "Apart from the questions within the section above, there are the following comments:\n- order of presentation:\n  - Algorithm 1: this is presented before equation (4) and (5), so it is hard to understand without having seen these equations.\n  - line 167: Proposition 3.1 mentioned two subsections before it is introduced\n- Proposition 3.1: is there a restriction on $C\\leq 0$? Otherwise your propostion does not provide any useful information\n- line 208: \"at test time, ...\". This is unclear. Please explain.\n- line 210\" Do you fine tune on data where you test or do you split the test data for fine tuning and for evaluation? \n- Line 225: What is LEADS, CoDA-l1/2? How can they be combined with your method?\n- Line 243: how can your method outperform the oracle?\n- Table 1: is the first colum in LEADS / CoDA-l1/2 the Train MSE or what metric is displayed there?\n\nSmaller comments:\n- line 85: ERM abbreviation not introduced\n- line 99: unclear what is $g$\n- Table 1 and 2: font size is too small. Keep same as in text preferrably.\n- Fig 3 is never referenced but fig 5 instead", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The method is simple and easy to follow with great foundations in clustering algorithms\n- The experiments show promising results\n- The discussion is insightful\n\nWeaknesses:\n- Presentation:\n  - In the abstract and introduction the problem formulation is unclear. You never mention the multi-environment setting. A motivating example with a clear figure would be of great value. \n  - Similarly, the environment variable is unclear. Is this some abstract latent variable or does it has physical meaning?\n  - The significance and importance of the problem should be worked out more clearly. Which practical settings does your method solve now.\n- Method:\n  - in line 155/156 you discuss that \"neural networks [...] effectively operates analogously to the \"centroid\" ...\". It would be important to show this observation in some form of experiment to have a strong basis for your method.\n  - For convergence speed your method seems to rely on a good initialisation. Please clarify its effect and which scheme you choose.\n- Experiments:\n  - It is unclear what LEADS and CoDA-l1/2 are doing and how it can be combined with your method.\n  - There are not baseline methods except for the basic environment variable assignments that your present. In the related work you present some baseline methods. A rigorous comparison would be necessary within the experiments", "was_revised": true, "final_justification": "The authors have addressed all my concerns. I do hope that my main concern about the presentation will be sufficiently adressed, which cannot be shown to all extend within the rebuttal phase.", "rebuttal": "We sincerely appreciate the reviewer’s time and insightful feedback on our work. Below, we address the raised concerns point by point. \n\n>  **[W1-a] In the abstract and introduction the problem formulation is unclear. You never mention the multi-environment setting. A motivating example with a clear figure would be of great value** \n\n**Response:**\n- We thank the reviewer for this valuable suggestion. We have now **clarified the multi-environment setting** in both the abstract and introduction, as in Lines 29-33:\n  - \"Recent work in dynamical systems addresses this by introducing a multi-environment setting, where trajectories follow distinct dynamics across environments. These studies developed generalization methods that learn a shared global component while accounting for environment-specific variations, avoiding the limitations of underperforming averaged models.\"\n- We have **added a new figure** to provide a motivating example in the revised manuscript.  \n\n>  **[W1-b] Similarly, the environment variable is unclear. Is this some abstract latent variable or does it has physical meaning?** \n\n**Response:**\n-  The environment variable (or environment label) in our work is an abstract latent variable that indicates which environment a given trajectory belongs to. \n-  Each environment has distinct underlying system dynamics, and both the dynamics and environment labels are unknown a priori—our method learns them jointly. \n- We **have clarified this** in the revised paper in Lines 36-37:\n  - \"These environment labels enable algorithms to identify and exploit both similarities and differences across environments.\"  \n\n>  **[W1-c] The significance and importance of the problem should be worked out more clearly. Which practical settings does your method solve now.** \n\n**Response:**\nWe sincerely thank the reviewer for raising this important point.  \n\n- **Theoretical & Practical Significance:** Our work addresses a fundamental challenge in machine learning—OOD generalization for dynamic systems—with broad theoretical and real-world implications. Unlike standard OOD settings that assume known environment labels, we focus on the more realistic yet understudied scenario where such labels are unavailable.  \n\n- **Key Motivations & Real-World Relevance:** The absence of environment labels in dynamic systems often arises due to inherent complexities, including:  \n  - **Data Collection Constraints:** In ecological studies, critical environmental factors (e.g., location and height) could be unrecorded.  \n  - **Multi-Source Data Aggregation:** In distributed sensor networks, data from heterogeneous sources often lack consistent environment annotations. \n  - **Privacy & Regulatory Barriers:** In healthcare or finance, sharing environment-specific metadata (e.g., patient demographics) is frequently restricted.  \n\nBy formalizing and solving this unlabeled OOD setting, our method bridges a critical gap in deploying robust dynamic systems in real-world applications. We **have expanded these points** in the revised manuscript to better emphasize both the novelty and applicability of our approach.  \n\n>  **[W2-a]  in line 155/156 you discuss that \"neural networks [...] effectively operates analogously to the \"centroid\" ...\". It would be important to show this observation in some form of experiment to have a strong basis for your method.** \n\n**Response:**\nWe thank the reviewer for this insightful suggestion. To empirically support our observation, we conducted an additional experiment measuring the MSE loss for each training sample across different neural networks (NNs). The results (samples shown below for brevity and **full table now included in the Appendix**) reveal two key findings:  \n1. **Each training sample has a uniquely best-fit NN** (i.e., one NN achieves minimal loss for that sample).  \n2. **This NN also generalizes to a cluster of trajectories**—specifically, those originating from the same environment.  \n\n|Traj.|NN1|NN2|NN3|NN4|NN5|NN6|NN7|NN8|NN9|Assign.|\n|---|---|---|---|---|---|---|---|---|---|---|\n|1|2.1E-1|1.2E-1|2.8E-1|7.8E-6|8.3E-2|1.6E-1|3.1E-1|2.5E-1|2.4E-1|3|\n|2|2.1E-1|1.2E-1|2.7E-1|9.4E-6|8.4E-2|1.5E-1|3.4E-1|2.4E-1|2.3E-1|3|\n|3|2.1E-1|1.2E-1|2.6E-1|1.0E-5|8.4E-2|1.5E-1|3.3E-1|2.4E-1|2.3E-1|3|\n|4|2.5E-1|1.5E-1|3.8E-1|2.0E-5|1.1E-1|1.9E-1|4.0E-1|3.1E-1|3.1E-1|3|\n|5|3.3E-2|1.6E-1|1.1E-1|8.3E-2|5.4E-6|5.4E-2|3.7E-1|7.3E-2|1.4E-1|4|\n|6|3.2E-2|1.9E-1|1.1E-1|8.5E-2|4.3E-6|5.6E-2|4.2E-1|6.8E-2|1.4E-1|4|\n|7|3.1E-2|1.9E-1|1.1E-1|8.5E-2|4.5E-6|5.5E-2|4.2E-1|6.7E-2|1.4E-1|4|\n|8|2.8E-2|2.7E-1|2.1E-1|1.1E-1|2.2E-5|9.6E-2|5.4E-1|1.2E-1|2.4E-1|4|\n|...|...|...|...|...|...|...|...|...|...|...|\n|29|1.9E-2|1.8E-1|1.0E-2|2.5E-1|7.2E-2|2.0E-2|3.2E-1|2.5E-6|5.6E-2|7|\n|30|2.1E-2|2.0E-1|1.1E-2|2.4E-1|6.8E-2|2.1E-2|3.7E-1|2.9E-6|6.4E-2|7|\n|31|2.0E-2|2.0E-1|1.1E-2|2.4E-1|6.7E-2|2.1E-2|3.7E-1|3.0E-6|6.4E-2|7|\n|32|7.1E-2|2.6E-1|2.0E-2|3.1E-1|1.2E-1|2.7E-2|4.2E-1|8.7E-6|7.4E-2|7|\n|33|5.5E-2|1.6E-1|1.2E-6|2.8E-1|1.1E-1|1.9E-2|2.4E-1|1.0E-2|2.5E-2|2|\n|34|6.0E-2|1.7E-1|3.1E-6|2.7E-1|1.1E-1|1.8E-2|2.8E-1|1.1E-2|2.8E-2|2|\n|35|6.0E-2|1.6E-1|3.1E-6|2.7E-1|1.1E-1|1.7E-2|2.8E-1|1.1E-2|2.8E-2|2|\n|36|1.6E-1|2.3E-1|1.2E-5|3.8E-1|2.1E-1|3.9E-2|3.3E-1|2.1E-2|3.4E-2|2|\n\nThis alignment between \"centroid-like\" NN behavior and environment-specific trajectory clusters reinforces our methodological rationale. We have **added this analysis and extended our discussion** in the revised manuscript.  \n\n>  **[W2-b] For convergence speed your method seems to rely on a good initialisation. Please clarify its effect and which scheme you choose.** \n\n**Response:**\n- All NN params in our method are actually **randomly initialized**. The convergence speed differences observed in **Figure 3** (e.g., LV learning faster than GS) arise from inherent task complexity rather than initialization.\n- LV’s simpler dynamics lead to rapid convergence (e.g., stable assignments before epoch 50), even when initial assignments are random.  \n- We **have clarified this point in the revision** to avoid potential misunderstandings. \n\n>  **[W3-a] & [Q5] It is unclear what LEADS and CoDA-l1/2 are doing and how it can be combined with your method.** \n\n**Response:**\nOur method addresses **environment inference** for dynamic system generalization, which requires building upon an existing OOD generalization model for dynamic systems. We adopt LEADS and CoDA-l1/2 as our base OOD generalization models because they are well-established in this field.  We **have clarified this integration** in the revised manuscript.  \n\n\n>  **[W3-b] There are not baseline methods except for the basic environment variable assignments that your present. In the related work you present some baseline methods. A rigorous comparison would be necessary within the experiments.** \n\n**Response:**\n- Our work is the **first** to address **environment inference** for unlabeled trajectory data in dynamic system generalization. Since this problem has not been studied before, no direct assignment baselines exist for comparison. \n- While we discuss related methods for generalization in the related work, these are not designed for environment inference—a key distinction highlighted in our paper.  \n- For clarity, we have **explicitly emphasized our contribution** in the revised manuscript, distinguishing our problem setting from prior work. \n\n>  **[Q1] Order of presentation: 1) Algorithm 1 [...] & 2) Proposition 3.1** \n\n**Response:**\nWe appreciate the feedback. We **have reordered the presentation**. \n\n>  **[Q2] Proposition 3.1: is there a restriction on $c\\leq0$?** \n\n**Response:**\nThank you for catching this. The proposition requires $C > 0$ to ensure the loss decreases by a positive constant. We’ve corrected this in the revision.  \n\n>  **[Q3] line 208: \"at test time, ...\". This is unclear. Please explain.** \n\n**Response:**\nThank you for pointing this out. Instead of assuming test environment labels are known, we are dealing with a greater challenge that test label is also unknown. In this case, the label is inferred by comparing predicted vs. test trajectories over the first temporal interval. We **have now clarified** in the revised paper.\n\n>  **[Q4] line 210\" Do you fine tune on data where you test or do you split the test data for fine tuning and for evaluation?** \n\n**Response:**\nOur method follows standard domain adaptation practice that finetunes on the test domain data. We **have clarify this point** in the revision.\n\n>  **[Q6] Line 243: how can your method outperform the oracle?** \n\n**Response:**\nThank you for pointing this out, \nOur method can outperform the oracle due to three key factors:\n- Our optimized environment labels may better capture generalization patterns than predefined oracle labels\n- Real-world trajectory overlaps make rigid oracle assignments suboptimal. Certain trajectories may lie closer to clusters in other environments than their assigned oracle labels suggest.\n- By optimizing environment assignments during training, our method resolves label ambiguities adaptively, prioritizing generalizable dynamics over strict adherence to oracle label fidelity. \nWe **have now clarify this effect** in the revision.\n\n>  **[Q7] Table 1: is the first colum in LEADS / CoDA-l1/2 the Train MSE or what metric is displayed there?** \n\n**Response:** \nThe first column shows the training MSE. We **have clarified this** in the revised table header.\n\n>  **[Smaller comments] 1) line 85: ERM abbreviation; 2) line 99: unclear what is g; 3) Table 1 and 2: font size is too small. 4) Fig 3 is never referenced but fig 5 instead** \n\n**Response:** \n1. ERM is the expected risk minimization. We **have now defined** the ERM in the revision.\n2. $g$ stands for the neural network learning the individual part. we **have now clarified** $g$ in the revision.\n3. We **have adjusted the font size** in table 1 and 2.\n4. We **have fixed this latex reference mistake**.\n\nAgain, we sincerely appreciate your valuable comments and welcome further discussion."}]

Author rebuttal:

Meta-review: The authors discuss fitting a neural network to a dynamical system whose coefficients are drawn from a mixture of “environments”. Lacking direct label information about the environments, the authors propose a bi-level optimization to estimate latent environment indicators alongside the parameters for environment-specific dynamics. Authors were positive about the submission, praising the simplicity and clarity of the proposed methods, as well as the theoretical and empirical results. There were some reviewer concerns about the paper (e.g. scaling to many environments, assuming access to ground truth coefficients) which were assuaged during the rebuttal period. I recommending accepting the paper, and encourage the authors to integrate the reviewer feedback into the camera ready version.
Decision: Accept (spotlight)