# SS-ESOAP: Self-Scaled Adaptive Preconditioning for Physics-Informed Learning

arXiv: https://arxiv.org/abs/2608.29448 · HTML: https://mlsanity.com/paper/2608.29448/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 33 references that are in the map.
- First posted on arXiv 29 Aug 2026.

## Builds on (most cited first, 20 of 20)

- [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)
- [SOAP: Improving and Stabilizing Shampoo using Adam](https://mlsanity.com/paper/2409.11321/) (2024)
- [Challenges in Training PINNs: A Loss Landscape Perspective](https://mlsanity.com/paper/2402.01868/) (2024)
- [PirateNets: Physics-informed Deep Learning with Residual Adaptive Networks](https://mlsanity.com/paper/2402.00326/) (2024)
- [PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.11833/) (2023)
- [Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective](https://mlsanity.com/paper/2502.00604/) (2025)
- [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)
- [Gauss-Newton Natural Gradient Descent for Physics-Informed Computational Fluid Dynamics](https://mlsanity.com/paper/2402.10680/) (2024)
- [Purifying Shampoo: Investigating Shampoo's Heuristics by Decomposing its Preconditioner](https://mlsanity.com/paper/2506.03595/) (2025)
- [Kronecker-Factored Approximate Curvature for Physics-Informed Neural Networks](https://mlsanity.com/paper/2405.15603/) (2024)
- [Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks](https://mlsanity.com/paper/2201.06780/) (2022)
- [Preconditioning for Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.00531/) (2024)
- [FP64 is All You Need: Rethinking Failure Modes in Physics-Informed Neural Networks](https://mlsanity.com/paper/2505.10949/) (2025)
- [Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networks](https://mlsanity.com/paper/2604.05230/) (2026)
- [Achieving High Accuracy with PINNs via Energy Natural Gradients](https://mlsanity.com/paper/2302.13163/) (2023)
- [ANaGRAM: A Natural Gradient Relative to Adapted Model for efficient PINNs learning](https://mlsanity.com/paper/2412.10782/) (2024)
- [High precision PINNs in unbounded domains: application to singularity formulation in PDEs](https://mlsanity.com/paper/2506.19243/) (2025)
- [Dual Natural Gradient Descent for Scalable Training of Physics-Informed Neural Networks](https://mlsanity.com/paper/2505.21404/) (2025)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) often face ill-conditioned objectives that limit high-accuracy training. Dense quasi-Newton methods improve local conditioning but require expensive optimizer state, while Kronecker-factored methods such as SOAP scale to larger networks but rely on periodic basis updates. We introduce method, which augments SOAP-style preconditioning with a scalar secant-energy correction adapted to Kronecker geometry and an adaptive basis update followed by variance-state downscaling. We characterize the directional secant matching induced by the scalar correction and give a bound on variance-state mismatch across basis changes. Across eight PDE benchmarks, method attains the lowest final residual on six, including Burgers and Boussinesq, while SOAP-family baselines perform better on Gray-Scott and Ginzburg-Landau. On Boussinesq, method reaches a residual of 10-5 in 4.1 hours with 9.2 GB peak VRAM, while Adam does not reach this target within 14 hours. Three-seed L2 and H1 errors on four representative PDEs support the link between lower residuals and improved solution accuracy. These results position method as a scalable option for stiff, high-accuracy physics-informed training, rather than a uniform replacement for existing optimizers.