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Closed-form Solutions: A New Perspective on Solving Differential Equations
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Proposes SSDE, an unsupervised RL framework (RNN generator with risk-seeking policy gradients, RSCO, and recursive per-dimension exploration) that derives closed-form symbolic solutions to ODEs/PDEs guided by physics-based residual rewards.
Problem
Deriving interpretable closed-form analytical solutions to differential equations without expert mathematical knowledge, where prior symbolic/GP/PINN-based methods yield complex, approximate, or physically inconsistent solutions and scale poorly to high-dimensional PDEs.
Approach
Treat symbolic expression discovery as an MDP solved by an RNN with risk-seeking policy gradients, using physics-regularized rewards (PDE residual, boundary and initial condition errors) computed via automatic differentiation; add risk-seeking constant optimization that first fits BC/IC before refining the PDE residual, and a recursive scheme that solves one variable at a time as parametric expres
Key results
- SSDE achieves 100% complete recovery rate on all six HD-PDE benchmarks with L_PHY around 1e-5, while PINN+DSR, KAN and PR-GPSR reach 0% recovery.
- On the Gamma Poisson set, SSDE reaches 100% recovery on all 12 problems versus PINN+DSR which fails (0%) on several, e.g. Gamma4, Gamma7, Gamma9.
- RSCO gives a reported 4.65x speedup (934s vs 200s) in discovery time over the ablated variant.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SSDE instead of PR-GPSR | Closed-form solving of Poisson2D · Poisson2D | L_PHY: 7.20e-5±2.57e-6 vs. 1.90e-2±2.32e-3 | Helped |
| Use SSDE instead of PR-GPSR | Closed-form solving of Poisson3D · Poisson3D | L_PHY: 5.70e-5±1.33e-6 vs. 1.80e-1±1.76e-2 | Helped |
| Use SSDE instead of PR-GPSR | Closed-form solving of Heat2D · Heat2D | L_PHY: 1.47e-6±7.64e-7 vs. 2.15e-1±1.28e-2 | Helped |
| Use SSDE instead of PR-GPSR | Closed-form solving of Heat3D · Heat3D | L_PHY: 1.05e-5±1.23e-6 vs. 8.87e-1±2.96e-1 | Helped |
| Use SSDE instead of PR-GPSR | Closed-form solving of Wave2D · Wave2D | L_PHY: 4.25e-5±2.35e-6 vs. 4.24e-1±3.69e-1 | Helped |
| Use SSDE instead of PR-GPSR | Closed-form solving of Wave3D · Wave3D | L_PHY: 8.17e-6±1.56e-6 vs. 9.87e-1±2.46e-1 | Helped |
| Add RSCO constant optimization (ablation) | SSDE · PDE benchmarks | complete recovery rate | Helped |
| Add recursive exploration (RecurExp) (ablation) | SSDE · High-dimensional PDEs | complete recovery rate | Helped |
| Add RSCO (ablation) | SSDE · Solution discovery | speedup: 4.65× | Helped |
| Add RSCO (ablation) | SSDE · Solution discovery | expression evaluations: 22% more | Hurt |
Surprising findings
- Hard-constraint-first constant optimization (fit BC/IC before PDE residual) is reported to accelerate search while the authors also note RSCO requires about 22% more expression evaluations.
Limitations
- Recursive exploration relies on the solution being decomposable into per-variable parametric expressions, which may not hold generally.
- Evaluated only on relatively simple, polynomial-like or smooth benchmark PDEs with known closed forms.
- Compared against baselines whose hyperparameters and implementations are not detailed in the main text.
Experimental setup
Datasets: Poisson2D Poisson3D Heat2D Heat3D Wave2D Wave3D Gamma dataset (Poisson-type 1D/2D problems) Nguyen expressions
Benchmarks: Poisson2D Poisson3D Heat2D Heat3D Wave2D Wave3D Gamma dataset
Metrics: L_PHY (mean physics-constraint RMSE) P_RE (complete recovery rate over 20 runs) mean relative L2 error wall-clock time
Baselines: PINN+DSR KAN PR-GPSR Mathematica FEX DSR PINN
Builds on: deep symbolic regression risk-seeking policy gradient PINNs DSR FEX
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