Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Message Passing Neural PDE Solvers
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use MP-PDE instead of FNO-PF | E1 Burgers shock modeling · E1 | accumulated error: 0.54 -> 1.55; 0.51 -> 1.67; 0.57 -> 1.47 | Hurt |
| Use MP-PDE instead of FNO-PF | generalization across PDE coefficients · E2 and E3 | accumulated error: E2: 2.53 -> 1.58; 2.27 -> 1.63; 2.38 -> 1.45. E3: 5.69 -> 4.26; 5.39 -> 3.74; 5.98 -> 3.70 | Helped |
| Add PDE coefficient features (ablation) | MP-PDE · E2 | accumulated error: 1.62 -> 1.58; 1.71 -> 1.63; 1.49 -> 1.45 | Helped |
| Add PDE coefficient and boundary-condition features (ablation) | MP-PDE · WE3 mixed boundary conditions | MSE: 38.775 -> 0.097; 20.445 -> 0.106; 16.859 -> 0.219; 17.591 -> 0.379 | Helped |
| Train FNO with temporal bundling and pushforward rather than unrolled training (ablation) | FNO-RNN · E1 | accumulated error: 11.93 -> 0.54; 29.98 -> 0.51; 10.44 -> 0.57 | Helped |
| Apply the pushforward trick (ablation) | autoregressive solver · E1 long rollouts | survival time | Helped |
| Replace pushforward perturbations with Gaussian noise (σ = 0.01) (ablation) | autoregressive solver · E1 long rollouts | survival time | Hurt |
| Add Gaussian perturbations (ablation) | autoregressive solver · E1 rollout accuracy | accuracy | Hurt |
| Stop gradients through the first unrolled step (ablation) | pushforward training · autoregressive PDE rollouts | stability | Helped |
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