Tensor Gaussian Processes: Efficient Solvers for Nonlinear PDEs
This work addresses the problem of solving nonlinear PDEs more efficiently for researchers and practitioners in computational science, representing an incremental improvement by combining tensor decomposition with Gaussian processes.
The paper tackles the inefficiency and scalability issues of machine learning solvers for nonlinear PDEs by proposing TGPS, a tensor-Gaussian process solver that reduces computational complexity and enables scalability to massive collocation sets, achieving superior accuracy and efficiency in experiments on benchmark PDEs.
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 models 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.