# DeePoly: A High-Order Accuracy Scientific Machine Learning Framework for Function Approximation and Solving PDEs

arXiv: https://arxiv.org/abs/2506.04613 · HTML: https://mlsanity.com/paper/2506.04613/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 34 references that are in the map.
- First posted on arXiv 5 Jun 2025.

## Builds on (most cited first, 9 of 9)

- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [The Loss Surfaces of Multilayer Networks](https://mlsanity.com/paper/1412.0233/) (2014)
- [Deep Learning without Poor Local Minima](https://mlsanity.com/paper/1605.07110/) (2016)
- [A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks](https://mlsanity.com/paper/2406.02917/) (2024)
- [Optimization for deep learning: theory and algorithms](https://mlsanity.com/paper/1912.08957/) (2019)
- [Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective](https://mlsanity.com/paper/2502.00604/) (2025)
- [Multi-stage Neural Networks: Function Approximator of Machine Precision](https://mlsanity.com/paper/2307.08934/) (2023)
- [Towards Understanding the Condensation of Neural Networks at Initial Training](https://mlsanity.com/paper/2105.11686/) (2021)

## Abstract (arXiv)

Recently, machine learning methods have gained significant traction in scientific computing, particularly for solving Partial Differential Equations (PDEs). However, methods based on deep neural networks (DNNs) often lack convergence guarantees and computational efficiency compared to traditional numerical schemes. This work introduces DeePoly, a novel framework that transforms the solution paradigm from pure non-convex parameter optimization to a two-stage approach: first employing a DNN to capture complex global features, followed by linear space optimization with combined DNN-extracted features (Spotter) and polynomial basis functions (Sniper). This strategic combination leverages the complementary strengths of both methods -- DNNs excel at approximating complex global features (i.e., high-gradient features) and stabilize the polynomial approximation while polynomial bases provide high-precision local corrections with convergence guarantees. Theoretical analysis and numerical experiments demonstrate that this approach significantly enhances both high-order accuracy and efficiency across diverse problem types while maintaining mesh-free and scheme-free properties. This paper also serves as a theoretical exposition for the open-source project DeePoly.