# Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces

arXiv: https://arxiv.org/abs/2301.04887 · HTML: https://mlsanity.com/paper/2301.04887/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 42 references that are in the map.
- First posted on arXiv 12 Jan 2023.

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

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Inverse-Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2107.00940/) (2021)
- [Accelerating Finite-temperature Kohn-Sham Density Functional Theory with Deep Neural Networks](https://mlsanity.com/paper/2010.04905/) (2020)
- [Replacing Automatic Differentiation by Sobolev Cubatures fastens Physics Informed Neural Nets and strengthens their Approximation Power](https://mlsanity.com/paper/2211.15443/) (2022)

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

- [Ensuring Topological Data-Structure Preservation under Autoencoder Compression due to Latent Space Regularization in Gauss--Legendre nodes](https://mlsanity.com/paper/2309.08228/) (2023)

## Abstract (arXiv)

We introduce a novel spectral, finite-dimensional approximation of general Sobolev spaces in terms of Chebyshev polynomials. Based on this polynomial surrogate model (PSM), we realise a variational formulation, solving a vast class of linear and non-linear partial differential equations (PDEs). The PSMs are as flexible as the physics-informed neural nets (PINNs) and provide an alternative for addressing inverse PDE problems, such as PDE-parameter inference. In contrast to PINNs, the PSMs result in a convex optimisation problem for a vast class of PDEs, including all linear ones, in which case the PSM-approximate is efficiently computable due to the exponential convergence rate of the underlying variational gradient descent. As a practical consequence prominent PDE problems were resolved by the PSMs without High Performance Computing (HPC) on a local machine. This gain in efficiency is complemented by an increase of approximation power, outperforming PINN alternatives in both accuracy and runtime. Beyond the empirical evidence we give here, the translation of classic PDE theory in terms of the Sobolev space approximates suggests the PSMs to be universally applicable to well-posed, regular forward and inverse PDE problems.