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

翻译：偏微分方程（PDE）的机器学习求解方法日益引起关注。然而，现有大多数方法（如神经网络求解器）依赖随机训练，效率低下且通常需要大量训练轮次。高斯过程（GP）/基于核的求解器虽具有数学原理基础，但在处理高维或挑战性偏微分方程所需的大量配置点时面临可扩展性问题。为克服这些限制，我们提出TGPS——一种基于张量高斯过程的求解器，该方法沿每个输入维度引入一维高斯过程的因子函数，并通过张量分解将其结合以逼近完整解。该设计将问题简化为学习一组一维高斯过程，大幅降低计算复杂度，并实现海量配置点集的可扩展性。为实现高效的非线性偏微分方程求解，我们采用部分冻结策略与牛顿法对非线性项进行线性化处理，进而发展出具备闭形式更新的交替最小二乘（ALS）方法，显著提升训练效率。我们为模型表达能力建立理论保证，并在标准正则性假设下给出收敛性证明与误差分析。在多个基准偏微分方程上的实验表明，与现有方法相比，本方法在准确性和效率方面均达到更优性能。代码已开源至https://github.com/BayesianAIGroup/TGPSolve-NonLinear-PDEs