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Environment Inference for Learning Generalizable Dynamical System

Shixuan Liu, Yue He, Haotian Wang, Wenjing Yang, Yunfei Wang, Peng Cui, Zhong Liu

TL;DR

The paper tackles learning generalizable dynamical systems when environment labels are unavailable by introducing DynaInfer, which infers latent environment assignments from trajectory prediction losses using fixed networks and alternates between environment assignment and parameter updates. It provides theoretical convergence guarantees for the alternating optimization and demonstrates strong performance on Lotka–Volterra, Gray–Scott, and Navier–Stokes benchmarks, often matching or surpassing Oracle-label baselines. The approach enables robust generalization across diverse environments without explicit domain annotations and shows favorable adaptation to unseen environments. Overall, DynaInfer offers a principled, scalable method for unlabeled environment inference in dynamical systems, with practical implications for scientific modeling and domain-generalization tasks.

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

TL;DR

The paper tackles learning generalizable dynamical systems when environment labels are unavailable by introducing DynaInfer, which infers latent environment assignments from trajectory prediction losses using fixed networks and alternates between environment assignment and parameter updates. It provides theoretical convergence guarantees for the alternating optimization and demonstrates strong performance on Lotka–Volterra, Gray–Scott, and Navier–Stokes benchmarks, often matching or surpassing Oracle-label baselines. The approach enables robust generalization across diverse environments without explicit domain annotations and shows favorable adaptation to unseen environments. Overall, DynaInfer offers a principled, scalable method for unlabeled environment inference in dynamical systems, with practical implications for scientific modeling and domain-generalization tasks.

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.
Paper Structure (32 sections, 1 theorem, 25 equations, 13 figures, 10 tables, 1 algorithm)

This paper contains 32 sections, 1 theorem, 25 equations, 13 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Dynamical Systems
  4. Multi-Environment Dynamical Systems Learning
  5. Environment Inference for Multi-Environment Learning
  6. The DynaInfer Framework
  7. Bias-aware Environment Assignment
  8. Assignment-driven Optimization
  9. Theoretical Property
  10. Experiments
  11. Environment Specification
  12. Experimental Setting and Baselines
  13. Experimental Results
  14. In-domain Generalization Results
  15. Domain Adaptation Results
  16. ...and 17 more sections

Key Result

Proposition 3.1

For all rounds $1 \le r < T_r$, we must have Furthermore, suppose the space of $\mathop{\mathrm{arg\,min}}\limits_{\theta, \boldsymbol{\phi}}R_{\hat{\boldsymbol{e}}}(\theta, \boldsymbol{\phi})$ is finite for all $\hat{\boldsymbol{e}} \in [M]^N$. Then there exists a constant $C>0$ such that if $r > 1$ and $R_{\hat{\boldsymbol{e}}^{(r + 1)}}(\th

Figures (13)

  • Figure 1: Environment labels, required by current generalization methods, are often unavailable.
  • Figure 2: Model Framework.
  • Figure 3: Left: Predicted test trajectories from 3 environments vs. ground truth and Oracle for LV with 2 base models. Middle and Right: Predicted last 3 states for GS and NS, respectively, vs. ground truth and Oracle using 2 base models. Each environment is shown by row. For CoDA, we use the best Oracle result's CoDA version to save space. See Appendix \ref{['appendix_plot_dynamic']} for all the visualizations.
  • Figure 4: Environment assignment probability over time, averaged over 5 runs, with LEADS as base model (on LV (top) and NS (bottom); see Appendix \ref{['sec:fig_conv_gs']} for GS). The assignment converges to the true label faster than the designated training steps. A similar trend is observed with the CoDA model.
  • Figure 5: Performance across assumed number of environments with DynaInfer (log-scale y-axis). The peak performance aligns with the true number of environments (bold on x-axis) with high probability, and remains stable thereafter.
  • ...and 8 more figures

Theorems & Definitions (3)

  • Proposition 3.1
  • Remark 3.1
  • proof