GRAFT-ATHENA projects combinatorial method choices into factored trees that embed as fingerprints in a metric space, enabling an agentic system to accumulate experience across domains and autonomously discover new numerical techniques for physics-informed problems.
Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networks
5 Pith papers cite this work. Polarity classification is still indexing.
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abstract
Efficient and robust optimization is essential for neural networks, enabling scientific machine learning models to converge rapidly to very high accuracy -- faithfully capturing complex physical behavior governed by differential equations. In this work, we present advanced optimization strategies to accelerate the convergence of physics-informed neural networks (PINNs) for challenging partial (PDEs) and ordinary differential equations (ODEs). Specifically, we provide efficient implementations of the Natural Gradient (NG) optimizer, Self-Scaling BFGS and Broyden optimizers, and demonstrate their performance on problems including the Helmholtz equation, Stokes flow, inviscid Burgers equation, Euler equations for high-speed flows, and stiff ODEs arising in pharmacokinetics and pharmacodynamics. Beyond optimizer development, we also propose new PINN-based methods for solving the inviscid Burgers and Euler equations, and compare the resulting solutions against high-order numerical methods to provide a rigorous and fair assessment. Finally, we address the challenge of scaling these quasi-Newton optimizers for batched training, enabling efficient and scalable solutions for large data-driven problems.
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2026 5verdicts
UNVERDICTED 5representative citing papers
Regularized Newton's method for neural networks converges exponentially to zero loss with uniform spectral rates in the infinite-width limit via a derived Newton neural tangent kernel.
A PINN solves the time-dependent quasi-static MHD equations in axisymmetric tokamak geometry without training data and reproduces vertical plasma displacement seen in ground-truth simulations.
A secant-based adaptive correction augments first-order optimizers to improve convergence speed, stability, and accuracy when training PINNs on challenging PDEs.
A neural network augmented with the geometric mean of multiple small parameters approximates solutions to singularly perturbed dynamical systems with satisfactory accuracy on tested coupled cases.
citing papers explorer
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GRAFT-ATHENA: Self-Improving Agentic Teams for Autonomous Discovery and Evolutionary Numerical Algorithms
GRAFT-ATHENA projects combinatorial method choices into factored trees that embed as fingerprints in a metric space, enabling an agentic system to accumulate experience across domains and autonomously discover new numerical techniques for physics-informed problems.
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Convergence Analysis of Newton's Method for Neural Networks in the Overparameterized Limit
Regularized Newton's method for neural networks converges exponentially to zero loss with uniform spectral rates in the infinite-width limit via a derived Newton neural tangent kernel.
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A Physics-Informed Neural Network for Solving the Quasi-static Magnetohydrodynamic Equations
A PINN solves the time-dependent quasi-static MHD equations in axisymmetric tokamak geometry without training data and reproduces vertical plasma displacement seen in ground-truth simulations.
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Lightweight Geometric Adaptation for Training Physics-Informed Neural Networks
A secant-based adaptive correction augments first-order optimizers to improve convergence speed, stability, and accuracy when training PINNs on challenging PDEs.
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Two-scale Neural Networks for Singularly Perturbed Dynamical Systems with Multiple Parameters
A neural network augmented with the geometric mean of multiple small parameters approximates solutions to singularly perturbed dynamical systems with satisfactory accuracy on tested coupled cases.