# Lightweight Geometric Adaptation for Training Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2604.15392 · HTML: https://mlsanity.com/paper/2604.15392/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 47 references that are in the map.
- First posted on arXiv 16 Apr 2026.

## Builds on (most cited first, 30 of 31)

- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [ASAM: Adaptive Sharpness-Aware Minimization for Scale-Invariant Learning of Deep Neural Networks](https://mlsanity.com/paper/2102.11600/) (2021)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [Optimization, Learning, and Games with Predictable Sequences](https://mlsanity.com/paper/1311.1869/) (2013)
- [Adan: Adaptive Nesterov Momentum Algorithm for Faster Optimizing Deep Models](https://mlsanity.com/paper/2208.06677/) (2022)
- [A Simple Convergence Proof of Adam and Adagrad](https://mlsanity.com/paper/2003.02395/) (2020)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [A Unified Analysis of Extra-gradient and Optimistic Gradient Methods for Saddle Point Problems: Proximal Point Approach](https://mlsanity.com/paper/1901.08511/) (2019)
- [Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile](https://mlsanity.com/paper/1807.02629/) (2018)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Challenges in Training PINNs: A Loss Landscape Perspective](https://mlsanity.com/paper/2402.01868/) (2024)
- [The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization](https://mlsanity.com/paper/1807.03907/) (2018)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks](https://mlsanity.com/paper/1905.01205/) (2019)
- [ASGO: Adaptive Structured Gradient Optimization](https://mlsanity.com/paper/2503.20762/) (2025)
- [Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective](https://mlsanity.com/paper/2502.00604/) (2025)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [A Novel Convergence Analysis for Algorithms of the Adam Family](https://mlsanity.com/paper/2112.03459/) (2021)
- [Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2501.16371/) (2025)
- [Unveiling the optimization process of Physics Informed Neural Networks: How accurate and competitive can PINNs be?](https://mlsanity.com/paper/2405.04230/) (2024)
- [Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networks](https://mlsanity.com/paper/2604.05230/) (2026)
- [Initialization-enhanced Physics-Informed Neural Network with Domain Decomposition (IDPINN)](https://mlsanity.com/paper/2406.03172/) (2024)
- [Binary structured physics-informed neural networks for solving equations with rapidly changing solutions](https://mlsanity.com/paper/2401.12806/) (2024)
- [Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspective](https://mlsanity.com/paper/2506.19805/) (2025)
- [A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learning](https://mlsanity.com/paper/2509.14198/) (2025)

## Built on by (most cited first, 1 of 1)

- [From Non-Convex Self-Concordant Regularization to Scalable Quasi-Newton Training of PINNs](https://mlsanity.com/paper/2608.04206/) (2026)

## Abstract (arXiv)

Physics-Informed Neural Networks (PINNs) often suffer from slow convergence, training instability, and reduced accuracy on challenging partial differential equations due to the anisotropic and rapidly varying geometry of their loss landscapes. We propose a lightweight curvature-aware optimization framework that augments existing first-order optimizers with an adaptive predictive correction based on secant information. Consecutive gradient differences are used as a cheap proxy for local geometric change, together with a step-normalized secant curvature indicator to control the correction strength. The framework is plug-and-play, computationally efficient, and broadly compatible with existing optimizers, without explicitly forming second-order matrices. Experiments on diverse PDE benchmarks show consistent improvements in convergence speed, training stability, and solution accuracy over standard optimizers and strong baselines, including on the high-dimensional heat equation, Gray--Scott system, Belousov--Zhabotinsky system, and 2D Kuramoto--Sivashinsky system.