RANDSMAPs: Random-Feature/multi-Scale Neural Decoders with Mass Preservation

TL;DR

RANDSMAPs are explainable neural decoders that incorporate conservation laws, leveraging random features and multi-scale analysis to improve manifold learning tasks with high accuracy and mass preservation.

Contribution

This paper introduces RANDSMAPs, a novel neural decoder framework that explicitly enforces mass preservation and captures multi-scale structures, with theoretical foundations and practical validation.

Findings

High reconstruction accuracy on benchmark problems

Maintains mass conservation at machine precision

Low computational cost compared to existing methods

Abstract

We introduce RANDSMAPs (Random-feature/multi-scale neural decoders with Mass Preservation), numerical analysis-informed, explainable neural decoders designed to explicitly respect conservation laws when solving the challenging ill-posed pre-image problem in manifold learning. We start by proving the equivalence of vanilla random Fourier feature neural networks to Radial Basis Function interpolation and the double Diffusion Maps (based on Geometric Harmonics) decoders in the deterministic limit. We then establish the theoretical foundations for RANDSMAP and introduce its multiscale variant to capture structures across multiple scales. We formulate and derive the closed-form solution of the corresponding constrained optimization problem and prove the mass preservation property. Numerically, we assess the performance of RANDSMAP on three benchmark problems/datasets with mass preservation obtained by the Lighthill-Whitham-Richards traffic flow PDE with shock waves, 2D rotated MRI brain images, and the Hughes crowd dynamics PDEs. We demonstrate that RANDSMAPs yield high reconstruction accuracy at low computational cost and maintain mass conservation at single-machine precision. In its vanilla formulation, the scheme remains applicable to the classical pre-image problem, i.e., when mass-preservation constraints are not imposed.