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Interpretable Solutions for Multi-Physics PDEs Using T-NNGP
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Proposes T-NNGP, a hybrid neural-guided genetic programming method that combines FEM numerical solutions, PhySO-style RNN deep RL candidate generation, and a decoupling-guided GP to find interpretable symbolic solutions for coupled PDE systems.
Problem
Obtaining interpretable symbolic solutions for systems of coupled PDEs in multiphysics simulation, where existing symbolic regression methods only handle single PDEs and numerical methods give only numerical outputs.
Approach
Combines FEM-generated physical datasets, RNN-based deep RL candidate expression generation (PhySO) into a neural-guided knowledge base, and a universal decoupling strategy that orders unknown functions by constraint degree to guide genetic programming crossover and mutation.
Key results
- T-NNGP achieves lower PDE-residual MSE than PhySO in all 9 instances, e.g., Joule heating 2 at 1.10e-13 vs 2.85e14 for PhySO.
- On numerical data fitting (MAE), T-NNGP achieves the best result in only 1 case and the same order of magnitude as the best in 8 results.
- Ablation on diffusion-convection shows decoupling strategy and update mechanism lower mean population fitness over 100 generations.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use T-NNGP instead of PhySO to generate PDE solution expressions | Diffusion-convection · Diffusion-convection | PDE residual MSE: 1.60e-02 -> 4.02e-03 | Helped |
| Use T-NNGP instead of PhySO to generate PDE solution expressions | T-NNGP · Joule heating | PDE residual MSE: 1.11e15 -> 2.36e00 | Helped |
| Use T-NNGP instead of PhySO to generate PDE solution expressions | T-NNGP · Parabolic | PDE residual MSE: 3.35e02 -> 1.93e00 | Helped |
| Add the decoupling strategy (ablation) | T-NNGP · Diffusion-convection | mean population fitness | Helped |
| Add the update mechanism (ablation) | T-NNGP · Diffusion-convection | mean population fitness | Helped |
Surprising findings
- T-NNGP does not win on MAE against DeepM&Mnet and PhySO on numerical data fitting, with the authors attributing this to PDE constraints and approximate numerical training data.
Limitations
- Symbolic regression limited to numerical physics datasets rather than real-world observations.
- MAE on numerical datasets is not strong relative to baselines; MAE does not fully measure PDE solution accuracy.
- Evaluated on only three PDE systems.
Experimental setup
Datasets: FEM-generated numerical solutions for diffusion-convection system FEM-generated numerical solutions for Joule heating system FEM-generated numerical solutions for parabolic system
Benchmarks: Diffusion-convection PDE system Joule heating PDE system Parabolic PDE system
Metrics: MSE of PDE residual improvement rate (IR) MAE variance of error mean fitness convergence
Baselines: PhySO DeepM&Mnet
Builds on: PhySO DeepM&Mnet symbolic regression with deep reinforcement learning
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