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Derivative-enhanced Deep Operator Network
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Proposes DE-DeepONet, a DeepONet trained with output-to-input derivative (Fréchet/Gateaux) labels on a linearly reduced (ASM or KLE) input basis, and shows derivative loss also helps FNO.
Problem
DeepONet surrogates for parametric PDEs approximate solutions well but their input-to-output derivatives are inaccurate, and high-dimensional parameter inputs make derivative training expensive, especially with limited training data.
Approach
Combine linear input dimension reduction (active subspace method using derivatives, or KLE) with a derivative loss term on the branch-net input-output Jacobian; derivative labels are computed matrix-free via linearized PDE solves, and the derivative loss is also applied to FNO.
Key results
- DE-DeepONet markedly reduces solution and derivative errors versus vanilla DeepONet across all training sizes, per the reported error figures.
- DE-FNO achieves the best accuracy among all models when training samples are 256 or 1024, at much higher training cost (0.215 s/iter vs 0.015 s/iter for FNO).
- ASM input basis gives lower output reconstruction error than KLE with 16 bases (Navier-Stokes 5.8% vs 17.4%; hyperelasticity 2.7% vs 3.8%).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add derivative loss to DeepONet training | DeepONet · Hyperelasticity and Navier--Stokes | relative solution and derivative errors | Helped |
| add derivative loss to FNO training | FNO · Hyperelasticity and Navier--Stokes, 256 or 1024 training samples | relative solution and derivative errors | Helped |
| use ASM instead of KLE for input dimension reduction (ablation) | DE-DeepONet · Hyperelasticity | relative L2 output reconstruction error: 3.8% -> 2.7% | Helped |
| use ASM instead of KLE for input dimension reduction (ablation) | DE-DeepONet · Navier--Stokes | relative L2 output reconstruction error: 17.4% -> 5.8% | Helped |
| use ASM instead of KLE for input dimension reduction (ablation) | DE-DeepONet · Hyperelasticity and Navier--Stokes | solution and derivative approximation errors | Helped |
| use full inputs instead of reduced inputs | FNO compared with DE-DeepONet and DINO · Hyperelasticity and Navier--Stokes, 1024 training samples | approximation errors | Helped |
| add derivative loss | FNO · Hyperelasticity and Navier--Stokes | training time per iteration | Hurt |
Surprising findings
- FNO with full inputs beats DE-DeepONet and DINO when training data is large (1024 samples), attributed to the linear input reduction error that cannot be removed by more data.
Limitations
- Requires derivative information (adjoint/linearized PDE solves) during training, which may be unavailable for legacy codes or unknown PDE forms.
- Input dimension reduction may be inaccurate or infeasible for intrinsically high-dimensional problems such as high-frequency wave equations.
- Derivative loss increases training time substantially, e.g., DE-FNO is about 14x slower per iteration than FNO.
Experimental setup
Datasets: synthetic hyperelasticity PDE data (FEniCS, 64x64 mesh) synthetic Navier-Stokes lid-driven cavity data (FEniCS, 64x64 mesh)
Benchmarks: hyperelasticity Navier-Stokes lid-driven cavity
Metrics: relative L2 error relative H1 error relative Frobenius error of derivative wall-clock training time inference time
Baselines: DeepONet FNO DINO DeepONet (ASM) DeepONet (KLE) DE-FNO
Builds on: DeepONet Fourier neural operator derivative-informed neural operator (DINO) active subspace method self-adaptive learning rate annealing
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