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Environment Inference for Learning Generalizable Dynamical System
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Proposes DynaInfer, which infers unlabeled environment assignments for dynamical system trajectories by alternating K-means-style assignment (lowest prediction loss under fixed networks) with network retraining, and shows it beats non-oracle labeling and often matches or beats oracle labels on LV, GS, and NS systems.
Problem
Multi-environment generalization for learning dynamical systems requires environment labels that are often unavailable in practice (data acquisition limits, privacy, lost labels in public datasets). The paper asks how to infer environment assignments from unlabeled mixed trajectories while learning
Approach
Treats each trained neural network as a K-means-style centroid: each trajectory is assigned to the environment whose network yields the lowest regression loss, and networks are then retrained on the inferred assignments. A proposition shows the loss is non-increasing across rounds and decreases by a constant when the assignment changes.
Key results
- On in-domain LV with LEADS, DynaInfer achieves test MSE 7.93e-5 versus 7.02e-5 for Oracle and 4.91e-4 for One per Env, while All in One is 7.41e-2.
- On in-domain NS with LEADS, DynaInfer test MSE is 7.05e-3 versus 6.55e-3 for Oracle and 4.11e-2 for One per Env.
- In adaptation on GS with LEADS, DynaInfer test MSE is 1.00e-4 versus 2.21e-4 for Oracle, and assignments converge to true labels faster than the training steps.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| infer environment labels (DynaInfer) instead of All in One grouping | LEADS on LV/GS/NS in-domain · in-domain LV, GS, NS | Test MSE: e.g. LV 7.93E-5 vs 7.41E-2 | Helped |
| DynaInfer environment inference vs. true labels (Oracle) | LEADS on GS and NS in-domain · in-domain GS, NS | Test MSE: GS 4.14E-5 vs 1.34E-4 (better); NS 7.05E-3 vs 6.55E-3 (worse) | Mixed |
| infer environment labels (DynaInfer) instead of One per Env grouping | LEADS on LV in-domain · in-domain LV | Test MSE: 7.93E-5 vs 4.91E-4 | Helped |
| random environment assignment instead of learned assignment | CoDA-l1 and CoDA-l2 in adaptation · domain adaptation LV, GS, NS | Test MSE: Random slightly diminishes CoDA performance across all datasets | Hurt |
| DynaInfer environment inference vs. Oracle in domain adaptation | LEADS in adaptation · domain adaptation LV, GS, NS | Test MSE: LV 5.77E-4 vs 1.67E-3 (Oracle); GS 1.00E-4 vs 2.21E-4 | Helped |
| increase assumed number of environments M beyond true value (ablation) | DynaInfer on LV · LV, M sweep 1 to 36 | Test MSE and learned label count: learned labels stay in 8-11 range for M from 5 to 36 (true M=9) | No clear effect |
| underestimating M (fewer environments than true) (ablation) | DynaInfer on LV · LV, M sweep | Test MSE: M=1 gives 7.41E-2 vs 7.93E-5 at M=9 | Hurt |
Surprising findings
- DynaInfer matches or exceeds Oracle (true environment labels) performance on several in-domain and adaptation settings, notably GS and NS.
- Test MSE in the in-domain GS setting is lower for DynaInfer than Oracle with LEADS on training loss (3.60e-5 vs 7.73e-5), suggesting inferred labels can outperform ground-truth partitions.
Limitations
- MSE is inadequate for chaotic systems, which the authors note as a limitation and future direction.
- Boundary conditions and complex domain geometries are not well handled by current learning methods.
- Requires an assumed number of environments M; performance peaks at the true M, and over-estimation requires incremental search.
Experimental setup
Datasets: Lotka-Volterra simulated trajectories Gray-Scott simulated trajectories Navier-Stokes simulated trajectories robot motion trajectory dataset
Benchmarks: Lotka-Volterra Gray-Scott Navier-Stokes robot motion
Metrics: MSE MAPE
Baselines: All in One One per Env Random Oracle LEADS CoDA-l1 CoDA-l2
Builds on: LEADS CoDA K-means
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