# Tensor Gaussian Processes: Efficient Solvers for Nonlinear PDEs

arXiv: https://arxiv.org/abs/2510.13772 · HTML: https://mlsanity.com/paper/2510.13772/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 49 references that are in the map.
- First posted on arXiv 15 Oct 2025.

## Builds on (most cited first, 22 of 22)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP)](https://mlsanity.com/paper/1503.01057/) (2015)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Deep Learning of Vortex Induced Vibrations](https://mlsanity.com/paper/1808.08952/) (2018)
- [A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions](https://mlsanity.com/paper/2302.14227/) (2023)
- [Sparse Cholesky factorization by Kullback-Leibler minimization](https://mlsanity.com/paper/2004.14455/) (2020)
- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [Scalable Gaussian Processes with Billions of Inducing Inputs via Tensor Train Decomposition](https://mlsanity.com/paper/1710.07324/) (2017)
- [Thoughts on Massively Scalable Gaussian Processes](https://mlsanity.com/paper/1511.01870/) (2015)
- [Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs](https://mlsanity.com/paper/2305.04962/) (2023)
- [AutoIP: A United Framework to Integrate Physics into Gaussian Processes](https://mlsanity.com/paper/2202.12316/) (2022)
- [Solving high-dimensional parabolic PDEs using the tensor train format](https://mlsanity.com/paper/2102.11830/) (2021)
- [Solving High Frequency and Multi-Scale PDEs with Gaussian Processes](https://mlsanity.com/paper/2311.04465/) (2023)
- [Toward Efficient Kernel-Based Solvers for Nonlinear PDEs](https://mlsanity.com/paper/2410.11165/) (2024)
- [Approximating optimal feedback controllers of finite horizon control problems using hierarchical tensor formats](https://mlsanity.com/paper/2104.06108/) (2021)
- [Meta Learning of Interface Conditions for Multi-Domain Physics-Informed Neural Networks](https://mlsanity.com/paper/2210.12669/) (2022)
- [Approximative Policy Iteration for Exit Time Feedback Control Problems driven by Stochastic Differential Equations using Tensor Train format](https://mlsanity.com/paper/2010.04465/) (2020)

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

- [Accelerated Learning of High Dimensional Functions with a Tensor-Featured Training Network](https://mlsanity.com/paper/2608.10351/) (2026)

## Abstract (arXiv)

Machine learning solvers for partial differential equations (PDEs) have attracted growing interest. However, most existing approaches, such as neural network solvers, rely on stochastic training, which is inefficient and typically requires a great many training epochs. Gaussian process (GP)/kernel-based solvers, while mathematical principled, suffer from scalability issues when handling large numbers of collocation points often needed for challenging or higher-dimensional PDEs. To overcome these limitations, we propose TGPS, a tensor-GP-based solver that introduces factor functions along each input dimension using one-dimensional GPs and combines them via tensor decomposition to approximate the full solution. This design reduces the task to learning a collection of one-dimensional GPs, substantially lowering computational complexity, and enabling scalability to massive collocation sets. For efficient nonlinear PDE solving, we use a partial freezing strategy and Newton's method to linerize the nonlinear terms. We then develop an alternating least squares (ALS) approach that admits closed-form updates, thereby substantially enhancing the training efficiency. We establish theoretical guarantees on the expressivity of our model, together with convergence proof and error analysis under standard regularity assumptions. Experiments on several benchmark PDEs demonstrate that our method achieves superior accuracy and efficiency compared to existing approaches. The code is released at https://github.com/BayesianAIGroup/TGPSolve-NonLinear-PDEs