Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Symmetry-Informed Governing Equation Discovery
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Proposes leveraging Lie point symmetries of autonomous ODEs to constrain or regularize equation discovery (sparse regression and genetic programming), improving success probability and noise robustness.
Problem
Data-driven equation discovery for ODEs searches an unnecessarily large space and often yields inaccurate or overly complex equations because it ignores physical symmetries such as frame invariance.
Approach
Derives equivariance constraints of the flow map from time-independent Lie point symmetries of ODEs; for linear symmetries it solves the constraints exactly to parametrize a reduced SINDy coefficient space, and for general symmetries it adds a relative infinitesimal symmetry loss as regularization; unknown symmetries are learned with LaLiGAN.
Key results
- On the damped oscillator at 20% noise, Equiv SINDy-c reaches 0.90 joint success probability versus 0.15 for SINDy and 0.00 for GP and D-CODE.
- On Lotka-Volterra at 99% noise, Equiv GP-r reaches 0.8 joint success probability versus 0.0 for GP and D-CODE.
- On the glycolytic oscillator at 20% noise, Equiv SINDy-r reaches 0.28 joint success versus 0.14 for SINDy.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| solve the linear rotation-symmetry constraint | SINDy · Damped oscillator | success probability (All): 0.15 -> 0.90 | Helped |
| solve the linear scaling-symmetry constraint | SINDy · Growth | success probability (All): 0.03 -> 1.00 | Helped |
| add learned-symmetry regularization | SINDy · Lotka-Volterra | success probability (All): 0.24 -> 0.36 | Helped |
| add learned-symmetry regularization | GP · Lotka-Volterra | success probability (All): 0.0 -> 0.8 | Helped |
| add learned-symmetry regularization | SINDy · Glycolytic oscillator | success probability (All): 0.14 -> 0.28 | Helped |
| add learned-symmetry regularization | GP · Glycolytic oscillator | success probability (All): 0.0 -> 0.1 | Helped |
| solve the symmetry constraint | SINDy · SEIR | success probability (Joint): 0.00 -> 0.14 | Helped |
Surprising findings
- WSINDy has very accurate parameter estimates once it discovers the correct form, though its success probability is low.
- D-CODE achieves the best long-horizon prediction on the glycolytic oscillator even when its discovered equation form is incorrect.
Limitations
- Symmetry regularization requires integrating learned equations or computing higher-order derivatives, adding computational cost and numerical error.
- Method is restricted to time-independent symmetries of autonomous ODEs.
- Unknown symmetries must be learned with LaLiGAN, whose accuracy affects downstream discovery.
Experimental setup
Datasets: simulated damped oscillator simulated growth system simulated Lotka-Volterra simulated glycolytic oscillator simulated SEIR simulated lambda-omega reaction-diffusion
Benchmarks: damped oscillator growth system Lotka-Volterra glycolytic oscillator SEIR
Metrics: success probability RMSE long-term prediction error
Baselines: SINDy WSINDy GP (PySR) D-CODE
Builds on: SINDy SINDy autoencoder LaLiGAN PySR Weak SINDy
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