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Boosting Generalization in Parametric PDE Neural Solvers through Adaptive Conditioning
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Proposes GEPS, a first-order, low-rank adaptive conditioning meta-learning method that adapts a small context vector per environment so neural PDE solvers generalize to unseen initial conditions, coefficients, forcings and domains.
Problem
Neural surrogate solvers for parametric PDEs generalize poorly across varying PDE parameters/environments under standard ERM training, especially with scarce adaptation data (one trajectory per new environment).
Approach
Learn shared network weights plus a low-dimensional per-environment context vector injected via low-rank modulation (A diag(c) B), trained with a first-order scheme without inner loops; at test time only the context is adapted from one trajectory, and it can be used in both purely data-driven and physics-hybrid (differentiable solver + NN) solvers.
Key results
- On the data-driven benchmarks GEPS achieves the best out-of-distribution error on Gray-Scott (1.86e-2 vs CoDA 2.89e-2) and Burgers (52.9e-3 vs LEADS 64.1e-3), with the lowest error on all four datasets among data-driven methods in the table.
- In the hybrid setting GEPS-Phy beats Phys-Ad and APHYNITY on Pendulum, Burgers and Kolmogorov out-of-distribution, but APHYNITY is best on Gray-Scott.
- On a 1200-environment combined PDE dataset, GEPS with 4M parameters achieves out-of-distribution relative L2 of 8.29e-2 versus 1.78e-1 for CoDA with 35M parameters.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use adaptive conditioning with GEPS instead of non-adaptive ERM | neural PDE solvers · Gray-Scott and Burgers, in-distribution and out-of-distribution | relative L2 loss | Helped |
| Use GEPS instead of CoDA | Combined PDE · Combined In-d and Out-d | relative L2 loss: 9.32e-2 -> 7.52e-3; 1.78e-1 -> 8.29e-2 | Helped |
| Use physics-aware GEPS-Phy instead of the strongest hybrid baseline | hybrid PDE solver · Pendulum and Burgers | relative L2 loss | Helped |
| Use GEPS-Phy rather than APHYNITY | hybrid PDE solver · Gray-Scott | relative L2 loss: 0.14\,$\pm$\,0.04 -> 0.67\,$\pm$\,0.05; 0.20\,$\pm$\,0.01 -> 0.83\,$\pm$\,0.05 | Hurt |
| Increase context-adaptation trajectories from 1 to 16 (ablation) | GEPS context adaptation · Kolmogorov flow | relative MSE loss: 5.22e-1 -> 5.20e-1 | No clear effect |
| Increase low-rank adaptation trajectories from 1 to 16 (ablation) | GEPS low-rank adaptation · Kolmogorov flow | relative MSE loss: 4.5e-1 -> 3.64e-1 | Helped |
| Initialize adaptation parameters orthogonally rather than with LoRA init (ablation) | GEPS · Gray-Scott | relative MSE loss: 2.35e-1 -> 2.22e-2 | Helped |
| Initialize adaptation parameters orthogonally rather than with LoRA init (ablation) | GEPS · Burgers | relative MSE loss: 4.95e-1 -> 1.46e-2 | Helped |
| Increase temporal history window size (ablation) | GEPS and ERM baselines · Burgers and Gray-Scott, in-distribution and out-of-distribution | Relative L2 loss | Helped |
| Use adaptive conditioning rather than ERM for temporal conditioning | neural PDE solvers · Burgers and Gray-Scott out-of-distribution | Relative L2 loss | Helped |
Surprising findings
- Classical Gaussian (LoRA) initialization of adaptation parameters performed worst; orthogonal initialization was best.
- Context-based adaptation (CoDA, CAVIA) did not scale with the number of adaptation trajectories, and meta-learning performed poorly on Kolmogorov flow in one-shot adaptation.
Limitations
- Adaptation still requires sufficient training environments and trajectories; one-shot adaptation performs poorly on complex dynamics like Kolmogorov flow.
- Experiments are on simple simulated dynamics; generalization to complex real-world conditions is not established.
- Accuracy of physical parameter estimation depends heavily on initialization.
Experimental setup
Datasets: damped driven pendulum (simulated) 2D Gray-Scott (simulated) 1D LES Burgers (simulated) 2D LES Kolmogorov flow (simulated) combined PDE (simulated)
Benchmarks: damped driven pendulum Gray-Scott Burgers LES Kolmogorov flow combined PDE
Metrics: relative L2 loss relative MSE
Baselines: CAVIA FOCA LEADS CoDA APHYNITY Phys-Ad FNO Transolver MP-PDE Poseidon
Builds on: CoDA LoRA APHYNITY Neural ODE FOCA
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