aiqbench analysis
MiNO advances scientific machine learning by shifting focus from direct field approximation to modeling the underlying propagator in phase space, offering superior handling of discontinuities and sharp fronts. This approach leverages microlocal analysis principles where smooth geometric rules generate nonsmooth solutions, effectively separating trainable error from frequency truncation limits. The method demonstrates significant efficiency gains over standard physics-informed neural networks on advection benchmarks while maintaining a compact model architecture.
MiNO: Cotangent-bundle propagator learning for PDEs
arXiv:2608.15187v1 Announce Type: new 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.
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.
Machine Learning (cs.LG); Computational Engineering, Finance, and Science (cs.CE); Numerical Analysis (math.NA)
arXiv:2608.15187 [cs.LG]
(or arXiv:2608.15187v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2608.15187 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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