Pith. sign in

REVIEW 2 cited by

TENG: Time-Evolving Natural Gradient for Solving PDEs With Deep Neural Nets Toward Machine Precision

Not yet reviewed by Pith; the record is open.

Re-run · record.json Download PDF Read on arXiv ↗

This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.

SPECIMEN: schema-true, not a live event

T0 review · schema-true

One-sentence machine reading of the paper's core claim.

pith:XXXXXXXX · record.json · timestamp

arxiv 2404.10771 v2 pith:BXBPGBB4 submitted 2024-04-16 cs.LG cs.NAmath.NAphysics.comp-ph

Zhuo Chen , Jacob McCarran , Esteban Vizcaino , Marin Soljačić , Di Luo This is my paper · ORCID
classification cs.LGcs.NAmath.NAphysics.comp-ph
keywords equationgradientnaturalpdesprecisiontengaccuracymachine
verification ladder T0 review T1 audit T2 compute T3 formal
0 comments
abstract

Partial differential equations (PDEs) are instrumental for modeling dynamical systems in science and engineering. The advent of neural networks has initiated a significant shift in tackling these complexities though challenges in accuracy persist, especially for initial value problems. In this paper, we introduce the $\textit{Time-Evolving Natural Gradient (TENG)}$, generalizing time-dependent variational principles and optimization-based time integration, leveraging natural gradient optimization to obtain high accuracy in neural-network-based PDE solutions. Our comprehensive development includes algorithms like TENG-Euler and its high-order variants, such as TENG-Heun, tailored for enhanced precision and efficiency. TENG's effectiveness is further validated through its performance, surpassing current leading methods and achieving $\textit{machine precision}$ in step-by-step optimizations across a spectrum of PDEs, including the heat equation, Allen-Cahn equation, and Burgers' equation.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Error analysis for learning the time-stepping operator of evolutionary PDEs

    math.NA 2025-09 conditional novelty 6.0 of 10

    For reaction-diffusion, forced parabolic, and viscous conservation law PDEs, neural networks can learn the numerical one-step time map with generalization error that is polynomial in mesh size, not exponential.

  2. BWLer: Barycentric Weight Layer Elucidates a Precision-Conditioning Tradeoff for PINNs

    cs.LG 2025-06 conditional novelty 6.0 of 10

    Adding a barycentric interpolation layer to PINNs lifts their precision ceiling, achieving up to 1e-13 relative error on smooth PDEs, while exposing a tradeoff between accuracy and loss conditioning.

Pith tools