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Sensitivity-Constrained Fourier Neural Operators for Forward and Inverse Problems in Parametric Differential Equations
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Adds a sensitivity (parameter-Jacobian) loss to Fourier Neural Operators, computed from differentiable or finite-difference solvers, to improve parameter inversion, sensitivity accuracy, data efficiency and robustness to input perturbation.
Problem
Neural operators such as FNO approximate solution paths of parametric differential equations well but give poor parameter sensitivities du/dp, which hurts parameter inversion, data efficiency and extrapolation under concept drift.
Approach
Augment the FNO data loss with a sensitivity loss that supervises the network's automatic-differentiation Jacobian with respect to physical parameters against Jacobians from a differentiable solver or finite differences, optionally combined with a PINN equation loss (SC-FNO-PINN).
Key results
- On single-parameter PDE1 inversion, SC-FNO reaches R²=0.998 versus 0.905 for FNO, with less than 1/5 the relative L2 inversion error of FNO and 1/4 that of FNO-PINN.
- At 40% parameter perturbation on PDE1, FNO's solution R² falls to 0.529 while SC-FNO keeps 0.912.
- On the 82-parameter zoned Burgers case with 100 samples, SC-FNO has relative L2 0.0087 for u versus 0.039 for FNO with 100 samples; the paper reports SC-FNO outperforms FNO with 500 samples in less than 1/3 the error.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add parameter-sensitivity loss | FNO · PDE1, parameter sensitivities on original parameter range | R²: 0.723 -> 0.924 | Helped |
| Add parameter-sensitivity loss | FNO · PDE2, parameter sensitivities on original parameter range | R²: 0.206 -> 0.987 | Helped |
| Add parameter-sensitivity loss | FNO · PDE1, solution at perturbed range (λ:0.4) | R²: 0.529 -> 0.912 | Helped |
| Add parameter-sensitivity loss | FNO · PDE1 multi-parameter inversion | R² (α): 0.642 -> 0.986 | Helped |
| Add parameter-sensitivity loss | FNO · ODE1 solution paths | R² (u): 0.996 -> 0.991 | Hurt |
| Add equation loss without sensitivity loss (ablation) | FNO · PDE2 parameter sensitivities | R²: 0.206 -> 0.137 | No clear effect |
| Add equation loss to SC-FNO (ablation) | SC-FNO · PDE1 and PDE2 solution and sensitivity metrics | Relative L²: 0.0175 -> 0.0112; 0.0761 -> 0.0577 | Mixed |
| Use FD-generated instead of AD-generated sensitivity targets (ablation) | SC-FNO · ODE1 solution paths | R²: 0.991 -> 0.968 | Hurt |
| Use FD-generated instead of AD-generated sensitivity targets (ablation) | SC-FNO · PDE1 solution paths | R²: 0.983 -> 0.963 | Hurt |
| Increase training samples from 500 to 1,000 | FNO · PDE1 training-data volume | R²: 0.8 -> around 0.9 | Helped |
| Add parameter-sensitivity loss | FNO · PDE2 zoned, 82 parameters, 500 samples | Relative L² (u): 0.0282 -> 0.0073 | Helped |
| Add parameter-sensitivity loss | FNO · PDE2 zoned, 82 parameters, 100 samples | Relative L² (u): 0.039 -> 0.0087 | Helped |
Surprising findings
- Adding the PINN equation loss to SC-FNO gives only minor gains in L2 accuracy, which the authors attribute to the absence of du/dp terms in the governing equations.
- FNO sensitivities show large unphysical oscillations even when its solution predictions are accurate.
Limitations
- Sensitivity supervision requires precomputed true Jacobians from a differentiable solver or finite differences, which adds dataset preparation cost.
- Gains from the PINN equation loss are small, and the paper does not evaluate ill-posed or non-differentiable physical models beyond finite-difference approximation.
Experimental setup
Datasets: synthetic ODE1 (composite harmonic oscillator) synthetic ODE2 (Duffing oscillator) synthetic PDE1 (damped wave) synthetic PDE2 (forced Burgers) synthetic PDE3 (Navier-Stokes vorticity) synthetic PDE4 (Allen-Cahn)
Benchmarks: synthetic ODE1 synthetic ODE2 synthetic PDE1 synthetic PDE2 synthetic PDE3 synthetic PDE4
Metrics: R² relative L2 error training time per epoch data-volume sensitivity
Baselines: FNO FNO-PINN WNO MWNO DeepONet
Builds on: Fourier Neural Operator physics-informed neural networks Wavelet Neural Operator Multiwavelet Neural Operator DeepONet
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