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Smooth Piecewise Cutting for Neural Operator to Handle Discontinuities and Sharp Transitions
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Proposes Cut-DeepONet, a two-stage framework that predicts input-dependent discontinuity locations with a Cutting Net and lifts the output into an augmented coordinate space where a DeepONet learns only smooth pieces.
Problem
Neural operators have continuous representations and struggle to learn solution operators with discontinuities and sharp transitions, requiring high capacity and high-resolution data; discontinuity locations are unknown for unseen inputs.
Approach
Lift the discontinuous output to a higher-dimensional space by assigning distinct region labels to subdomains separated by predicted discontinuities, so the operator only learns smooth pieces; a separate Cutting Net predicts discontinuity locations for unseen inputs, and the lifted solution is evaluated at predicted labels to yield true discontinuities.
Key results
- On linear advection, Cut-DeepONet reaches L1 0.009 and Dis 0.060 versus 0.025 / 0.167 for the best baseline (FNO).
- On inviscid Burgers', Cut-DeepONet reaches L1 0.010 vs 0.031 for HyperDeepONet, and Dis 0.053 vs 0.135.
- On the parsimonious sharp-transition model, Cut-DeepONet achieves the lowest L1 (2.032) but its Dis error (2.050) is higher than FNO (1.777).
Surprising findings
- The L1 and Dis metrics may underestimate Cut-DeepONet because a true discontinuity is double-counted, so standard metrics penalize sharp jumps.
- Cut-DeepONet's Dis error is worse than FNO on the parsimonious sharp-transition problem, attributed to random sampling underrepresenting near-transition regions.
Limitations
- Cutting Net assumes a fixed number of discontinuities; varying numbers are not handled.
- Cutting Net performance depends strongly on accuracy of the training discontinuity labels; solver errors propagate.
- Random sampling underrepresents near-discontinuity regions, hurting Dis error on sharp-transition problems.
Experimental setup
Datasets: linear advection simulations (generated) inviscid Burgers' equation simulations (Godunov solver, exact solution) parsimonious cardiomyocyte model simulations (generated)
Benchmarks: linear advection inviscid Burgers' equation parsimonious cardiomyocyte model
Metrics: L1 loss Dis loss (L1 near discontinuity) L2 error (Cutting Net)
Baselines: DeepONet FlexDeepONet HyperDeepONet FNO
Builds on: DeepONet Lifting the Winding Number FNO phi-DeepONet
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