🤖 AI Summary
This study addresses the regularity loss caused by discontinuities in PDE solving by proposing to learn phase-space propagators rather than directly fitting solution fields. By modeling phase and amplitude via eikonal and transport equations and reconstructing solutions through oscillatory integrals, the method transforms singularity propagation into a geometric problem, effectively decoupling it from field approximation. In discontinuous advection benchmarks, this approach rapidly converges to theoretical accuracy limits. Furthermore, it outperforms Fourier Neural Operators in smooth regimes and demonstrates robust generalization across multiple initial conditions with a single model. Consequently, this framework successfully overcomes the accuracy bottlenecks of traditional neural operators when handling non-smooth solutions.
📝 Abstract
Scientific machine learning for partial differential equations commonly targets solution fields, as in physics-informed neural networks, or solution maps, as in neural operators. We study a third target: the propagator itself, a phase and amplitude in phase space. The motivation is a gap in regularity. A transported discontinuity is nonsmooth in space and time, yet the rule that moves it can be a polynomial phase carrying unit amplitude, so the object that generates an evolution can be far smoother than the field it generates. The microlocal neural operator (MiNO) learns that object, using the eikonal equation for the phase and the transport equation for the amplitude, and recovers the solution by an oscillatory integral. Sharp fronts and caustics then belong to propagation geometry rather than to a field fitted pointwise. Small residuals certify more than the reconstructed field. They place the learned canonical relation, the geometry that carries singularities, close to the exact one, and they separate trainable error from the frequency-truncation tail. On a matched-budget discontinuous-advection benchmark, MiNO stops improving within 10,000 steps at the accuracy limit of its finite reconstruction window, a limit predicted in closed form, whereas a physics-informed neural network with neural-tangent-kernel loss balancing stays near its initial error. On smooth advection, the mean error is $3.84\times10^{-3}$ for MiNO and $3.12\times10^{-2}$ for a supervised Fourier neural operator. Single-branch MiNO is the smallest model compared, and one trained generator serves five unseen initial conditions without retraining.