Avy Soffer

Avy Soffer, Nguyen Gia Hien, Minh-Binh Tran

We introduce a microlocal phase-space-filtered physics-informed neural network (PINN--TDPSF or Microlocal PINNFilter) framework for wave propagation on unbounded domains. The method combines a slabwise neural residual approximation of the interior evolution with a time-dependent phase-space filter applied in a buffer surrounding the physical computational domain. The central idea is to replace local artificial-boundary penalties by a phase-space radiation mechanism: a component is removed only when it is localized near the artificial boundary and its group velocity points outward. The proposed method is not intended to replace FFT, spectral, or split-step solvers for known-coefficient forward problems where such methods are available and highly accurate. Instead, it embeds the time-dependent phase-space filter into a residual-based neural framework. This coupling is useful when open-domain wave propagation must be combined with nonlinear residuals, sparse or off-grid observations, unknown coefficients, variable interior media, or other non-FFT-diagonalizable physics. Numerical experiments for linear Schrödinger propagation, potential scattering, anisotropic Schrödinger dynamics, nonlinear Schrödinger wave packets, soliton stress tests, linearized Euler waves, and sparse-data recovery of a localized acoustic defect show that the method reduces artificial reflection and wraparound, uses group velocity correctly in anisotropic media, preserves physically incoming branch components, and provides diagnostics when the assumptions behind outgoing-packet filtering are violated.

Charles Siegel, Avy Soffer, Chris Stucchio

It has long been known how to construct radiation boundary conditions for the time dependent wave equation. Although arguments suggesting that they are accurate have been given, it is only recently that rigorous error bounds have been proved. Previous estimates show that the error caused by these methods behaves like epsilon C exp(gamma t) for any gamma > 0. We improve these results and show that the error behaves like epsilon t^2.

Avy Soffer, Chris Stucchio

For some anisotropic wave models, the PML (perfectly matched layer) method of open boundaries can become polynomially or exponentially unstable in time. In this work we present a new method of open boundaries, the phase space filter, which is stable for all wave equations. Outgoing waves can be characterized as waves located near the boundary of the computational domain with group velocities pointing outward. The phase space filtering algorithm consists of applying a filter to the solution that removes outgoing waves only. The method presented here is a simplified version of the original phase space filter, originally described in [22] for the Schrodinger equation. We apply this method to anisotropic wave models for which the PML is unstable, namely the Euler equations (linearized about a constant jet flow) and Maxwell's equations in an anisotropic medium. Stability of the phase space filter is proved.