Research map / Probabilistic and scientific modeling / Differential equations and PDEs
A Physics-preserved Transfer Learning Method for Differential Equations
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Proposes Physics-preserved Optimal Tensor Transport (POTT), a min-max optimal-transport-based transfer learning method that maps source product distributions toward the target and adds a physics regularizer, to correct distribution and operator bias for neural operators on DEs.
Problem
Neural operators for differential equations suffer performance degradation under domain shift (distribution bias and operator bias) between source and target, and existing transfer learning methods either lack generality or fail to preserve physics when the target has few samples.
Approach
Model the ideal transfer map as a physics-regularized optimal tensor transport map between source and target product distributions; the pushforward of source data is used together with few target samples to fine-tune the neural operator, with a dual (Lagrangian) min-max training scheme and a physical regularizer (or marginal-relation regularizer when no physics prior is available).
Key results
- On Darcy flow D2->D3 with 50 target samples, POTT reduces relative error from 0.4693 (finetuning) to 0.3527, and with 100 samples from 0.3553 to 0.2271.
- On Burgers' equation, POTT achieves average rMSE 0.1242 (50 samples) and 0.0809 (100 samples), versus 0.1532 and 0.0977 for finetuning.
- On cross-region ERA5 climate forecasting, POTT lowers latitude-weighted RMSE relative to finetuning (e.g., t2m at 6h: 2.65 vs 3.43) but remains above the ClimODE oracle trained on full target data.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use POTT instead of finetuning | FNO transfer on Burgers' equation · Burgers' equation average, 50 target samples | rMSE: 0.1532 -> 0.1242 | Helped |
| use POTT instead of finetuning | FNO transfer on Burgers' equation · Burgers' equation average, 100 target samples | rMSE: 0.0977 -> 0.0809 | Helped |
| use POTT instead of COD | DeepONet transfer on Darcy flow · Darcy flow average, 50 target samples | rMSE: 0.2444 -> 0.2095 | Helped |
| use POTT instead of DARE-GRAM | DeepONet transfer on Darcy flow · Darcy flow average, 100 target samples | rMSE: 0.1696 -> 0.1479 | Helped |
| use POTT instead of COD | DeepONet transfer on Advection equation · Advection equation average, 50 target samples | rMSE: 0.1458 -> 0.1032 | Helped |
| use POTT instead of COD | DeepONet transfer on Advection equation · Advection equation average, 100 target samples | rMSE: 0.0516 -> 0.0504 | Helped |
| use POTT instead of finetuning | ClimODE climate forecasting · ERA5 global forecasting, t2m, 6-hour lead | latitude-weighted RMSE: 3.43 ± 0.24 -> 2.65 ± 0.2 | Helped |
| use POTT instead of finetuning | ClimODE climate forecasting · ERA5 global forecasting, t2m, 36-hour lead | latitude-weighted RMSE: 3.86 ± 0.27 -> 3.13 ± 0.28 | Helped |
| add physical regularization to OTT (ablation) | POTT on Darcy flow · D1→D2 | physics preservation | Helped |
Surprising findings
- TL-DeepONet, DARE-GRAM and COD caused negative transfer on the simpler Burgers' D1->D3 task, performing worse than finetuning.
- Finetuning error maps show triangular patterns from the source domain distribution, indicating source-specific artifacts persist after finetuning.
Limitations
- Performance diminishes when target data is extremely small, limiting one-shot or few-shot use.
- Min-max optimization is complex and costly for large-scale models.
- Low data resolution in experiments; higher-resolution problems would need larger models.
Experimental setup
Datasets: Burgers' equation simulations Advection equation simulations Darcy flow simulations ERA5 (WeatherBench)
Benchmarks: Burgers' equation Advection equation Darcy flow ERA5 climate forecasting (WeatherBench)
Metrics: relative MSE (rMSE) latitude-weighted RMSE
Baselines: Src+Tgt Finetuning TL-DeepONet DARE-GRAM COD ClimaX
Builds on: Optimal transport (Monge/Kantorovich) neural optimal transport FNO DeepONet ClimODE
Closest papers by meaning
- PRDP: Progressively Refined Differentiable Physics (ICLR 2025 · similarity 0.87)
- Learning Data-Efficient and Generalizable Neural Operators via Fundamental Physics Knowledge (ICLR 2026 · similarity 0.87)
- Learning a Neural Solver for Parametric PDEs to Enhance Physics-Informed Methods (ICLR 2025 · similarity 0.86)
- An Interpretable Approach to the Solutions of High-Dimensional Partial Differential Equations (AAAI 2024 · similarity 0.85)
- Learning from Integral Losses in Physics Informed Neural Networks (ICML 2024 · similarity 0.85)
- Active Learning with Selective Time-Step Acquisition for PDEs (ICML 2025 · similarity 0.84)
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