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An Interpretable Approach to the Solutions of High-Dimensional Partial Differential Equations
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Proposes HD-TLGP, a genetic programming symbolic regression method for high-dimensional PDEs that transfers the structure of 1D analytical solutions, uses automatic differentiation for fitness, and adds pruning operators.
Problem
Finding fast, interpretable analytical solutions to high-dimensional PDEs, where genetic programming symbolic regression suffers from a very large search space and slow convergence.
Approach
A structural transfer mechanism builds a knowledge base by substituting the 1D analytical solution's structure into higher-dimensional variables and combining subtrees across dimensions, which seeds and guides GP search; automatic differentiation checks PDE and boundary residuals in the fitness function, and pruning operators control expression bloat.
Key results
- HD-TLGP achieved the best MSE on 7 of 12 PDE instances in Table 2 (versus 5 for FEM PDE solver).
- On advection 2, HD-TLGP reports MSE 1.85e-32 and recovers exact analytical forms such as sin-based solutions, while PR-GPSR gives approximate expressions.
- Convergence traces on advection 1 show HD-TLGP's best objective converges to a small value faster than PR-GPSR over 100 generations.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use HD-TLGP instead of PINN | PDE solving · heat1 | MSE: 6.19e-02±2.63e-02 -> 1.65e-06±8.22e-07 | Helped |
| Use HD-TLGP instead of PINN | PDE solving · heat2 | MSE: 2.22e-02±1.50e-02 -> 3.98e-09±2.84e-09 | Helped |
| Use HD-TLGP instead of PR-GPSR | PDE solving · heat3 | MSE: 1.70e-02±1.63e-02 -> 2.33e-08±8.40e-09 | Helped |
| Use HD-TLGP instead of PINN | PDE solving · heat4 | MSE: 4.59e-03±2.45e-03 -> 2.53e-08±1.48e-08 | Helped |
| Use HD-TLGP instead of PDE solver (FEM) | PDE solving · poisson1 | MSE: 9.86e-12±8.12e-12 -> 4.70e-08±4.67e-08 | Hurt |
| Use HD-TLGP instead of PDE solver (FEM) | PDE solving · poisson2 | MSE: 5.06e-09±4.80e-09 -> 1.33e-07±1.29e-07 | Hurt |
| Use HD-TLGP instead of PDE solver (FEM) | PDE solving · poisson3 | MSE: 9.27e-13±6.52e-13 -> 1.04e-08±2.84e-09 | Hurt |
| Use HD-TLGP instead of PDE solver (FEM) | PDE solving · poisson4 | MSE: 2.19e-11±1.09e-11 -> 1.53e-07±4.42e-08 | Hurt |
| Use HD-TLGP instead of PDE solver (FEM) | PDE solving · advection1 | MSE: 2.33e-11±6.99e-12 -> 3.5e-07±1.00e-07 | Hurt |
| Use HD-TLGP instead of PDE solver (FEM) | PDE solving · advection2 | MSE: 1.15e-06±1.15e-06 -> 1.85e-32±1.85e-32 | Helped |
| Use HD-TLGP instead of PINN | PDE solving · advection3 | MSE: 9.54e-03±1.26e-04 -> 4.75e-05±4.48e-05 | Helped |
| Use HD-TLGP instead of PINN | PDE solving · advection4 | MSE: 5.49e-02±2.67e-02 -> 00e-00±00e-00 | Helped |
Surprising findings
- The FEM PDE solver produced very large MSE and variance on advection 4 (2.60e+40), attributed to unstable numerical solutions and outliers.
Limitations
- Evaluated only on three PDE families (heat, Poisson, advection) with a limited set of instances.
- Some baseline solutions are omitted from Table 3 due to space, and some results are approximate fits rather than exact forms.
- Advection 4 and other results show high variance for baselines; the method's robustness across broader PDE types is not established.
Experimental setup
Datasets: synthetic observations from heat, Poisson, and advection PDEs (1D-to-high-dimensional analytical solutions)
Benchmarks: heat1-4 poisson1-4 advection1-4 (2D-4D PDE instances)
Metrics: mean squared error variance of error convergence of best objective per generation average individual length
Baselines: FEM (COMSOL PDE solver) PINN PR-GPSR
Builds on: PR-GPSR genetic programming symbolic regression automatic differentiation
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