Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Learning differentiable solvers for systems with hard constraints
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add a hard PDE-constrained layer instead of training with a soft PDE residual loss | FNO on 2D Darcy Flow · 2D Darcy Flow test set | relative L2 error: 3.86% ± 0.3% -> 1.82%± 0.04% | Helped |
| Add a hard PDE-constrained layer instead of training with a soft PDE residual loss | FNO on 2D Darcy Flow · 2D Darcy Flow test set | PDE residual test loss: 1.1355 ± 0.0433 -> 0.0457± 0.0021 | Helped |
| Add a hard PDE-constrained layer instead of training with a soft PDE residual loss | MLP on 1D Burgers · 1D Burgers test set | relative L2 error: 4.34% ± 0.33% -> 1.11 ± 0.11% | Helped |
| Add a hard PDE-constrained layer instead of training with a soft PDE residual loss | Dense network on 1D convection · 1D convection test set | relative L2 error: 2.59% ± 0.15% -> 1.32%± 0.02% | Helped |
| Add a hard PDE-constrained layer instead of training with a soft PDE residual loss | Dense network on 1D convection · 1D convection test set | PDE residual test loss: 774 ± 1.2 -> 9.84± 2.15 | Helped |
| Use learned basis functions instead of cubic interpolation from constrained points (ablation) | Trained PDE-CL on 1D convection · 1D convection 100x100 grid | accuracy: 36% | Helped |
| Use learned basis functions instead of cubic interpolation from constrained points (ablation) | Trained PDE-CL on 1D convection · 1D convection 1000x1000 grid | accuracy: 37% | Helped |
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