The Physical Cutoff Does Not Restore Homogenization: Phase-Dependent Burning in the Strain G-Equation

Michele Caprio

math.APAug 15, 2026

The Physical Cutoff Does Not Restore Homogenization: Phase-Dependent Burning in the Strain G-Equation

Authors: Michele Caprio

Abstract

We disprove the expectation stated by Xin, Yu, and Ronney that the physical positive part strain $G$ -equation should possess an effective burning velocity in cellular flows. For the standard cellular flow in dimension two $V_A(x_1,x_2)=A(-\sin x_1\cos x_2,\cos x_1\sin x_2)$ , if $0<d<20/399$ and $\sqrt{1+4d^2}<Ad\le1+d/10$ , then for every unit planar slope the periodic correction develops oscillations at least linearly in time. The solution remains bounded below on an explicit horizontal channel through $(π,0)$ , while at $(π/2,0)$ it decreases at rate at least $CA/\log A$ , with $C>0$ universal. The same conclusions hold for arbitrary continuous periodic perturbations of planar initial data. Under the physical scaling $V_A(x/\varepsilon)$ and $d_\varepsilon=\varepsilon d$ , an order one value gap persists between points at distance $O(\varepsilon)$ at every positive macroscopic time, so the rescaled solutions have no locally uniformly convergent subsequence. The proof uses the Hamiltonian sandwich $H_{\mathrm{unc}}\le H_+\le\widehat H$ . The upper comparator $\widehat H$ is a rectangular support function, equivalently an upper expectation over a state-dependent credal set, whose reversed control dynamics possess an invariant comparison channel. We also prove that for any $C^2$ incompressible periodic flow, every $\varepsilon$ -outward barrier certificate has covering radius at most $2d\varepsilon$ for all sufficiently small $\varepsilon$ . We further discuss implications for statistics and machine learning: rectangular, time-consistent local uncertainty need not imply forgetting of the initial state in the long run, so additional global stability or ergodicity conditions are needed in robust sequential decision making. Two Lean 4 appendices record conditional formalizations of a sufficient $p=e_1$ subregime and of the logical assembly of the rigidity theorem for barrier certificates.

Figures & tables

Figure 1. Schematic of the physical strain $G$ -equation ( 2 ). The zero level set $\Gamma_{t}$ represents the flame front, with unit normal $\nu=DG/|DG|$ . The surrounding flow $V$ carries the front, while the normal strain $\nu^{\mathsf{T}}S_{V}\nu$ modifies its local burning speed. The positive part cutoff suppresses burning under sufficiently strong compression without allowing reversal.

May 21, 2026

ARC-STAR: Auditable Post-Hoc Correction for PDE Foundation Models

Partial differential equation (PDE) foundation models are pretrained networks that forecast how physical fields like velocity and pressure evolve from a single reusable solver. On unfamiliar flows their predictions drift step by step, errors concentrate in a few regions, yet retraining destabilizes the network and uniform post-hoc correction overlooks this spatial concentration. To address this, we propose a frozen-solver post-hoc correction framework, Adaptive Risk-Calibrated Spatial Triage for Auditable Refinement (ARC-STAR). ARC-STAR organizes correction into three stages: a global corrector removes broad solver bias, a blockwise local refiner cleans the post-global residual, and, at deployment, a label-free score routes refinement to high-risk blocks under a compute budget. The framework is designed to be (i) frozen-host, preserving the pretrained solver without fine-tuning; (ii) auditable, with global and local stages trained and evaluated separately for measurable contributions; and (iii) budget-aware, using a blockwise interface that either refines the full field or routes limited compute to high-risk regions. Across five flow benchmarks spanning ten regime cells, ARC-STAR is the only method that cuts velocity rollout error by at least 36x over raw Poseidon on every cell. The global stage reduces raw host error by 91-99%, and the local stage further reduces the remaining post-global residual by up to 94.4%.

Chengze Li, Lingwei Wei, Li Sun +7

University of Illinois Chicago · 2Beijing University of Posts and Telecommunications · 3North China Electric Power University

PDE Surrogate Modeling PDE Foundation Models Computational Fluid Dynamics

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Jul 27, 2026

Score-Based Stabilization for Time-Dependent Problems

We propose a score-based stabilization framework for numerical simulation of partial differential equations, in which a learned score model defines a stabilization operator applied to provisional numerical updates. This operator augments standard time-stepping schemes by enforcing structure and physical consistency through a correction that drives iterates toward the manifold of admissible states. We show that the stabilization operator acts as a contraction toward this manifold, yielding a correction mechanism with basin-conditional stability. Numerical experiments on Advection, Korteweg-de Vries (KdV), Nonlinear Schrodinger (NLS), and Burgers' equations demonstrate improved robustness, suppression of nonphysical instabilities, and preservation of qualitative dynamics.

Eshed Gal, Eldad Haber, Uri Ascher

University of British Columbia Vancouver, British Columbia, Canada

Neural PDE Solvers PDE Solving Partial Differential Equations

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May 8, 2026

Exactness Matters for Physical Rule Enforcement

Autoregressive scientific forecasters often enforce physical or structural constraints by repairing each predicted state before feeding it back into the model. However, it remains unclear when stronger physical rule enforcement becomes reliable and when it becomes a source of distribution shift. We study this question through operator exactness, meaning whether the repair map is the identity on the target manifold and is aligned with the target geometry. We compare raw forecasting, post hoc repair, and in-loop repair across periodic incompressible Navier--Stokes, non-periodic CFDBench flows, and a hierarchical-forecasting support task. In the exact periodic regime, Fourier projection substantially improves rollout accuracy. On the NS-128 benchmark, a strong Raw-FNO has a final-step rollout MSE at horizon 100 of

(9.390 \pm 6.290)\times 10^{-5}

, and post hoc and in-loop projection reduce it to

(1.130 \pm 0.165)\times 10^{-6}

and

(5.370 \pm 0.113)\times 10^{-7}

. However, once an exact projection is unavailable and only approximate boundary-preserving cleanup is available, the ordering changes. Across cavity, tube, dam, and cylinder flow, stronger Poisson-based cleanup can reduce divergence while worsening rollout error; target-distortion MSE predicts this harm far better than a linear-system residual. Controlled mismatch, screened cleanup, adaptive gating, and external-backbone checks show that the best approximate-regime operating point can be raw or near-identity. Hierarchical forecasting gives the same broader pattern. Exact forecast reconciliation is a stable baseline, whereas blended top-down repair, a validation-tuned interpolation toward historical-proportion top-down reconciliation, is dataset-dependent. Thus, constraint enforcement should be benchmarked by operator--data alignment before enforcement strength.

Bum Jun Kim

The University of Tokyo, Japan

Time Series Forecasting Spatiotemporal Forecasting Computational Fluid Dynamics

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Abstract

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