Adaptive Spectral Trust Gate for Physics- Constrained Operator Learning

Abstract

Physics-informed machine learning improves the plausibility, data-efficiency and generalization of surrogate models by injecting prior physical knowledge into the learning process. The current approaches can be broadly divided into two main categories: soft constraints, which add a physics residual to the training loss but guarantee nothing at inference time, and hard constraints, which project the model output onto the constraint set exactly but apply the projection uniformly to every part of the signal — including parts that are dominated by noise, discretization error, or model mismatch, where the idealized physics is not actually trustworthy. This dissertation proposes the Adaptive Spectral Trust Gate (ASPINO), a mechanism that learns where to trust the physics. Operating in the Fourier domain on top of any surrogate model, a small gating network forms a per-mode convex combination of a data-driven soft path and a physics hard path. The gate is driven by features of the spectral coordinate and the spectral amplitude, so that it can apply the hard constraint in well-conditioned spectral regions and defer to the data-driven operator in regions corrupted by noise or aliasing. A single gate serves two very different hard paths - the linear Leray projection (incompressible flow) and a nonlinear rank-r SVD projection (massive-MIMO channel estimation). On the theoretical side, we give an empirical-Rademacher-complexity analysis: an unconditional safety floor — the gated class never exceeds the soft path it wraps — and, under a stated low-rank-transfer assumption, a capacity-reduction factor of 1 − ¯α (1 − √ρr), where ¯α is the fraction of capacity routed through the hard path. On Kolmogorov-flow denoising ASPINO is simultaneously the most accurate and near physical, dominating the unconstrained, hard and soft baselines; on ray-traced MIMO it improves a strong physics-informed baseline across all pilot budgets and SNRs without ever regressing. A third study, zero-shot super-resolution on the Poisson equation, confirms the discretization invariance of the gated construction. ASPINO is discretization-invariant and “plug-and-play” over the underlying operator.