Research map / Probabilistic and scientific modeling / Dynamical systems
Discovering Physics Laws of Dynamical Systems via Invariant Function Learning
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Introduces invariant function learning for ODE systems, proposing DIF, a causally-motivated encoder-decoder hypernetwork that disentangles invariant derivative functions from environment-specific ones via an MSE-based mutual information principle and adversarial environment discriminator.
Problem
Discovering shared invariant dynamics (physical laws) of ODE systems observed across multiple environments where environments differ in function forms, not only coefficients, without environment-specific mechanisms leaking into the discovered law.
Approach
Formulate invariant function learning as a causal graph in function space: parameterize derivative functions via a hypernetwork, maximize mutual information between the predicted invariant function and the full dynamics subject to independence from the environment (enforced adversarially), reducing to an MSE objective for practical training.
Key results
- On ME-Pendulum, the invariant function prediction reaches NRMSE 0.3561 on invariant trajectories, versus 0.7884 for the full function prediction.
- Over 75% of DIF's hyper-parameter candidates on ME-Pendulum outperform the best MAML and CoDA results, and more than 93.75% outperform the best IRM and VREx results.
- PySR recovers the frictionless pendulum invariant law as dω/dt ≈ -0.97 α² sin(θ), with dθ/dt ≈ 0.99 ω.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the invariant function rather than the full function for invariant trajectory prediction (ablation) | DIF · ME-Pendulum invariant trajectory (X^c) | NRMSE: 0.7884 -> 0.3561 | Helped |
| Use the invariant function rather than the full function for invariant trajectory prediction (ablation) | DIF · ME-Lotka-Volterra invariant trajectory (X^c) | NRMSE: 0.7919 -> 0.6194 | Helped |
| Use the invariant function rather than the full function for invariant trajectory prediction (ablation) | DIF · ME-SIREpidemic invariant trajectory (X^c) | NRMSE: 0.9867 -> 0.0652 | Helped |
| Use the reference-based hypernetwork implementation instead of the naïve implementation | DIF hypernetwork · Hypernetwork implementation | speedup: 16.8x | Helped |
| Use the reference-based hypernetwork implementation instead of the vectorized functional implementation | DIF hypernetwork · Hypernetwork implementation | speedup: 2x | Helped |
Surprising findings
- Symbolic regression on the extracted invariant function is imperfect on ME-SIREpidemic, with coefficient mismatches even though the invariant NRMSE is near-perfect.
- The authors note that effective invariant function learning does not always yield good PySR explanations.
Limitations
- Limited to ODE systems; PDE extension not addressed.
- Relies on symbolic regression (PySR) for interpretability, which struggles with many variables and samples.
- Only low-dimensional systems evaluated; high-dimensional graph-structured systems left to future work.
Experimental setup
Datasets: ME-Pendulum ME-Lotka-Volterra ME-SIREpidemic
Benchmarks: ME-Pendulum ME-Lotka-Volterra ME-SIREpidemic
Metrics: NRMSE symbolic regression explanation
Baselines: MAML CoDA IRM VREx
Builds on: invariant risk minimization CoDA MAML VREx PySR
Closest papers by meaning
- Symmetry-Informed Governing Equation Discovery (NeurIPS 2024 · similarity 0.89)
- Context-Informed Neural ODEs Unexpectedly Identify Broken Symmetries: Insights from the Poincaré–Hopf Theorem (ICML 2025 · similarity 0.87)
- CausalDynamics: A large‐scale benchmark for structural discovery of dynamical causal models (NeurIPS 2025 · similarity 0.87)
- Marrying Causal Representation Learning with Dynamical Systems for Science (NeurIPS 2024 · similarity 0.86)
- ODEFormer: Symbolic Regression of Dynamical Systems with Transformers (ICLR 2024 · similarity 0.86)
- Neural Context Flows for Meta-Learning of Dynamical Systems (ICLR 2025 · similarity 0.86)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.