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How to Re-enable PDE Loss for Physical Systems Modeling Under Partial Observation
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Proposes RPLPO, which re-enables PDE loss under partial observation by jointly training an encoder that reconstructs a learnable high-resolution state and a transition network, followed by semi-supervised two-stage fine-tuning on unlabeled data.
Problem
Physics-informed training with PDE loss needs high-resolution states, but sensor limits yield only partial observations, so finite-difference PDE residuals are biased and the generalization benefit of PDE loss is lost.
Approach
Encode recent partial observations into a learnable high-resolution state with a U-Net encoder, apply PDE loss between adjacent encoded high-resolution states and predicted transitions, then fine-tune transition on unlabeled data before refining the encoder with data loss.
Key results
- On five benchmarks, RPLPO achieves the lowest relative data loss; e.g., NSWE L_D 3.50E-2 vs best baseline FNO* 6.41E-2.
- RPLPO reduces reconstruction error epsilon on NSE to 2.76E-8 vs 1.85E-2 for FNO*-type baseline and improves Burgers epsilon to 1.85E-6.
- With 3x3 sparse observations, RPLPO L_D is 4.43E-2 vs FNO* 7.86E-2.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use RPLPO's learned high-resolution encoding and PDE-constrained transition instead of FNO* | physical-system prediction · Burgers, Wave, NSE, LSWE, NSWE | relative data loss: 1.68E-2 -> 1.37E-2 (Burgers); 3.58E-2 -> 2.64E-2 (Wave); 1.68E-2 -> 1.34E-2 (NSE); 4.75E-2 -> 2.44E-2 (LSWE); 6.41E-2 -> | Helped |
| Use RPLPO rather than the strongest baseline | physical-system reconstruction · Burgers, Wave, NSE, NSWE | reconstruction error ε: 9.80E-6 -> 1.85E-6 (Burgers); 1.18 -> 1.06 (Wave); 1.07E-2 -> 2.76E-8 (NSE); 3.16E-1 -> 2.03E-1 (NSWE) | Helped |
| Use RPLPO rather than PIDL | LSWE reconstruction · LSWE | reconstruction error ε: 1.38E-3 -> 1.47E-3 | Hurt |
| Add two-stage fine-tuning using unlabeled data (ablation) | RPLPO · NSWE, |D|=50 | relative data loss: 9.53E-2 -> 8.81E-2 | Helped |
| Add high-resolution labels for 1/3, 1/2, or all observations (ablation) | RPLPO · NSWE | relative data loss: 3.50E-2 -> 3.41E-2; 3.50E-2 -> 3.35E-2; 3.50E-2 -> 3.29E-2 | Helped |
| Add PDE loss terms to data-only training (ablation) | RPLPO · NSWE | relative data loss: 6.41E-2 -> 3.50E-2 | Helped |
| Replace the U-Net encoder with a Transformer (ablation) | RPLPO · NSWE | relative data loss: 3.50E-2 -> 4.70E-2 | Hurt |
| Replace the PDE-trained encoding module with nearest-neighbor, bilinear, or bicubic interpolation (ablation) | RPLPO · NSWE | reconstruction error ε: 2.03E-1 -> 8.40E-1 (nearest neighbors); 2.03E-1 -> 5.98E-1 (bilinear); 2.03E-1 -> 5.83E-1 (bicubic) | Hurt |
| Increase the number of recent observations from n=1 to n=4 (ablation) | RPLPO · NSWE | relative data loss: 8.94E-2 -> 3.50E-2 | Helped |
| Use more PDE-loss weighting (γ=1E-1 rather than γ=1E-3) (ablation) | RPLPO · NSWE | relative data loss: 4.16E-2 -> 3.50E-2 | Helped |
| Increase the PDE inaccuracy from accurate PDE to GRFs with std 10 (ablation) | RPLPO · NSWE, no data noise | relative data loss: 3.50E-2 -> 5.12E-2 | Hurt |
| Add Gaussian observation noise (ablation) | RPLPO with accurate PDE · NSWE | relative data loss: 3.50E-2 -> 5.05E-2 (10%); 3.50E-2 -> 6.11E-2 (20%); 3.50E-2 -> 6.80E-2 (30%) | Hurt |
Surprising findings
- PIDL (pure PDE loss, no data) achieves slightly better reconstruction error than RPLPO on LSWE, though its data loss is much larger.
- Using high-resolution labels for the encoder only modestly improves RPLPO (3.50E-2 to 3.29E-2 with all labels) while FNO* improves more.
Limitations
- Evaluated only on simulated PDE benchmarks with synthetic data at small resolutions (32x32 high-resolution state).
- Requires known PDE form and finite-difference schemes; inaccurate PDE is only tested with GRF perturbations and viscosity changes.
- Encoder uses fixed observation mesh, so zero-shot generalization to new meshes needs a different encoder (transformer with nearest-neighbor interpolation).
Experimental setup
Datasets: Synthetic Burgers, Wave, NSE, LSWE, NSWE trajectories generated from Gaussian random field initial conditions via RK4 an
Benchmarks: Burgers Wave NSE LSWE NSWE
Metrics: relative L2 data loss relative reconstruction error epsilon divergence error TKE error energy spectrum error
Baselines: PIDL LNPDE GNOT PeRCNN FNO FNO* PINO* MP-PDE MeshGraphNets CROM
Builds on: Fourier neural operator PINO PIDL
Closest papers by meaning
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