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Discovering Symmetries of ODEs by Symbolic Regression
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Adapts search-based symbolic regression (a multi-output DAG-based regressor) to find Lie point symmetry generators of ODE systems by minimizing a symmetry-condition loss on numerically simulated trajectories.
Problem
Finding Lie point symmetries of nonlinear ODE systems is hard for computer algebra systems (CASs) such as Mathematica's SYM package, which fail on many nonlinear systems and whose PDE-based determining equations often cannot be solved.
Approach
Use symbolic regression to directly search for the symmetry generator eta* (with the Hamiltonian/time-evolution part removed) by minimizing a loss derived from the symmetry condition evaluated on numerically integrated ODE solutions, with gradients via autodiff, rather than solving PDEs symbolically.
Key results
- SYM automatic mode fails on all ten test ODE systems, and SYM-Z (all but one generator component zero) succeeds on only one; SYM-S succeeds on 7 of 10.
- The symbolic regression approach finds symmetry generators for all ten problems, with numeric losses around 1e-13 to 1e-24.
- Generators are found in under six minutes on average (50 runs each) on a 64-core Xeon CPU, with runtime growing with generator expression size.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use symbolic regression to search for symmetry generators | Ten representative ODE systems · ODE1–ODE10 | symmetry-generator discovery | Helped |
| Use SYM automatic mode instead of symbolic regression | Ten representative ODE systems · ODE1–ODE10 | successful symmetry finding | Hurt |
| Use SYM-Z, setting all but one generator function to zero | Ten representative ODE systems · ODE1–ODE10 | successful symmetry finding | Hurt |
| Use SYM-S, setting all but one generator function to known-generator values | Ten representative ODE systems · ODE1–ODE10 | successful symmetry finding | Hurt |
Surprising findings
- Removing the Hamiltonian (time-evolution) component from generators makes multiple independent generators meaningful, enabling discovery of two independent symmetries for one nonlinear system.
Limitations
- Requires the ODE to be known in symbolic form and identifiability needs multiple time series (n+1 for n-dimensional systems).
- Runtime grows exponentially with the size of the expression DAG search space, so complex generators are expensive to find.
- Comparison is limited to ten hand-selected problems and only against Mathematica SYM.
Experimental setup
Datasets: numerically simulated ODE solutions via Runge-Kutta (RK45) for ten example ODE systems
Benchmarks: ten representative two-dimensional ODE systems (ODE1-ODE10) nonlinear independent-generators example system
Metrics: success/failure in finding symmetry generator numeric symmetry loss runtime
Baselines: Mathematica SYM package Maple symgen MathLie
Builds on: DAG-based symbolic regression framework Lie symmetry theory AI Feynman
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