PhyRes MDNF Physics Coupled Residual GNN Correction for Multilevel Discrete Neural Field Inversion

Jul 16, 2026

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Coefficient inversion under PDE constraints is ill conditioned because sparse observations weakly constrain fine-scale parameters, while single-resolution optimization must recover state and coefficient fields across all scales at once. This yields slow, initialization-sensitive convergence and risks approximation errors from learned transfer models that rely on offline datasets. PhyRes-MDNF introduces a fixed-physics multilevel discrete neural field framework with joint optimization across levels. Under the same fine-grid update budget, it cuts coefficient and state errors by about 85% and 90%, and improves KTC2023 EIT mean Otsu mIoU by ~3.4% over linearized CEM and 16.9% over direct single-level DNF.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages27

Reading Time1 h 8 min

Introduction

Challenges in coefficient inversion under PDE constraints

Existing approaches: finite methods, PINNs, and learned surrogates

PhyRes-MDNF for inverse problems

Discrete neural fields and fixed-physics optimization

Multilevel DNF backbone and level-to-level correction

Highlights PhyRes-MDNF: Physics-Coupled Residual GNN Correction for Multilevel Discrete Neural Field Inversion Zheng Lu, Jiwei Jia, Young Ju Lee • Multilevel DNF improves Darcy inversion under matched fine-grid updates. • PhyRes-GNN directly prolongs and corrects across levels without replacing physics. • LUNDIsim and KTC2023 demonstrate generality across inverse problems. arXiv :2607 .06237v1 [math .NA] 7 Jul 2026 PhyRes-MDNF: Physics-Coupled Residual GNN Correction for Multilevel Discrete Neural Field Inversion Zheng Lua , Jiwei Jiaa,c,∗ and Young Ju Leeb,∗∗ a School of Mathematics, Jilin University, Changchun, Jilin, 130012, China b Department of Mathematics, Texas State University, San Marcos, TX, 78666, USA cAI for Science and Engineering Center, Shenzhen Loop Area Institute, Shenzhen, 518048, China ARTICLE INFO Keywords: PDE-constrained inverse problems discrete neural fields multilevel optimization graph neural networks Darcy flow electrical impedance tomography AB STRACT Coefficient inversion on fine grids under PDE constraints is ill conditioned: sparse observations weakly constrain fine-scale parameters, and direct single-resolution optimization must recover state and coefficient fields across all scales simultaneously. This causes slow, initializationsensitive convergence; learned transfer models require offline data and can introduce approximation error into the numerical physics. We propose PhyRes-MDNF, a fixed-physics multilevel discrete neural field framework. On each level, a single-level DNF represents the inverse unknowns directly as trainable fields and optimizes the discrete objective. In the full-space Darcy realization, state fields 􀁕 and their shared coefficient field 􀁋 are optimized jointly in one fixed-physics inverse process. Between levels, one zero-initialized PhyRes-GNN jointly performs fixed-stencil prolongation and bounded residual correction to construct an incoming target representation, which a fixed initialization map converts to the next DNF variables. It is fitted anew from the observations and unchanged numerical model, without offline pretraining or fine-grid truth. Coarse levels therefore resolve large-scale structure before refined degrees of freedom are introduced, shortening the fine-grid optimization path while retaining the original discrete operator. Under the same final-grid update budget, the multilevel Darcy realization reduces coefficient and state errors by approximately 85% and 90%, respectively, demonstrating improved accuracy and final-grid iteration efficiency. On measured KTC2023 EIT data, the full-􀁗 pipeline improves mean Otsu mIoU by approximately 3.4% over the official linearized CEM reconstruction and 16.9% over direct single-level DNF. 1. Introduction Coefficient inversion for elliptic PDEs seeks a spatially varying material field from indirect measurements of states, voltages, or fluxes. The problem is intrinsically ill conditioned: observations are sparse relative to the number of coefficient degrees of freedom, fine-scale modes are only weakly visible, and errors in the reconstructed state can be absorbed by the coefficient. On a fine grid, direct single-resolution optimization must recover large-and smallscale structures simultaneously. This produces a long, initialization-sensitive optimization path and can cause the coefficient and state variables to settle into an inaccurate local solution before the large-scale structure has been identified [13, 30, 31] . Classical finite-difference and finite-element inverse methods retain a trusted discrete forward model, but optimizing all fine-grid inverse variables from the start remains expensive and poorly conditioned. Physics-informed neural networks (PINNs) instead represent states and coefficients with coordinate networks and fit their weights from observation and residual losses [16, 25] . Although flexible, this representation can introduce spectral bias, loss imbalanc

FAQs

Why is coefficient inversion for elliptic PDEs ill conditioned on fine grids?

Sparse observations provide weak constraints on fine-scale coefficient modes, and state reconstruction errors can be absorbed into the coefficient. As a result, direct single-resolution optimization must recover large- and small-scale structures simultaneously, making convergence long and sensitive to initialization.

What is PhyRes-MDNF and how does it differ from transfer or offline pretrained models?

PhyRes-MDNF is a fixed-physics multilevel discrete neural field framework that keeps the prescribed numerical operator unchanged. It is fitted anew from observations without offline pretraining or fine-grid truth, avoiding surrogate approximation errors being misinterpreted as physically plausible coefficient changes.

How does the method perform multilevel correction across resolutions?

Within the multilevel DNF backbone, one zero-initialized PhyRes-GNN performs fixed-stencil prolongation and bounded residual correction to build an incoming target representation for the next level. A fixed initialization map converts it to the next DNF variables, allowing coarse levels to resolve large-scale structure before introducing refined degrees of freedom.

Jul 16, 2026

Upload by Riley