Sensitivity-aware neural operator learning: from Navier–Stokes rollouts to flood and tsunamis
Pennsylvania State University
Abstract
Data-driven surrogates now approximate PDE solution maps orders of magnitude faster than solvers, yet the tasks we most need them for – parameter inversion, data assimilation, design and control, uncertainty quantification – consume the solver’s sensitivities, not its outputs. Operators trained to fit fields alone reproduce flows while getting ∂u/∂p wrong, then fail at those tasks and drift in long rollouts. Enforcing sensitivity is what makes a fast surrogate usable. Matching the solver’s Jacobian during training (SC-FNO, ICLR 2025) preserves field accuracy while enabling inversion over many parameters, with less data and shorter training, across equations and operator architectures. It also changes data scaling: with sensitivities enforced, information per sample grows with input dimensionality, blunting the curse of dimensionality for spatially distributed inputs and letting us reconstruct earthquake-induced seafloor deformation from sparse buoy records and forecast tsunami propagation with solver-level consistency in minutes. For autoregressive rollouts, refining each predicted window against the governing equation in its integral form, with the network frozen and no retraining, cuts error by 83% for turbulent Navier–Stokes, 40–72% for Burgers and 33–61% for the Richards equation of variably saturated flow. Because real domains are meshes and coastlines rather than periodic boxes, I will present spectral operators that learn on irregular geometries and transfer across them, and a multi-resolution shallow-water emulator that resolves wet–dry fronts while conserving volume under bathymetric uncertainty. Finally, gradient-faithful surrogates can stand in for solvers inside differentiable models, where neural closures are trained through the physics across thousands of catchments, recovering full-solver learning quality at roughly tenfold speedup.
About the speaker
Dr. Chaopeng Shen is Professor in Civil Engineering at The Pennsylvania State University. He received the Ph.D. degree in environmental engineering from Michigan State University, East Lansing, MI, USA, in 2009. His Ph.D. research focused on computational hydrology and he developed the hydrologic model Process-based Adaptive Watershed Simulator (PAWS), which was later coupled to the community land model to study the interactions between hydrology and ecosystem. He was a Post-Doctoral Research Associate with the Lawrence Berkeley National Laboratory, Berkeley, CA, USA, from 2011 to 2012, working on high-performance computational geophysics. His later efforts focused on harnessing the big data and machine learning (ML) and physics-informed ML opportunities in advancing hydrologic predictions and understanding. As an early advocate for ML in geosciences, he has written technical, editorial, review and collective opinion papers on hydrologic deep learning to call to attention the emerging opportunities for scientific advances. He promotes differentiable modeling, which seamlessly integrates neural networks and physics for knowledge discovery and is now being implemented by NOAA as a candidate for the next-generational National Water Model. In addition, his research interests also include floodplain systems, scaling issues, process-based hydrologic modeling, and hydrologic data mining. He is currently Editor of Water Resources Research, Chief Editor for Frontiers in AI: Water and AI, and previously an inaugural Editor of Journal of Geophysical Research: Machine Learning and Computation.
