Toward Efficient Kernel-Based Solvers for Nonlinear PDEs
Authors: Zhitong Xu, Da Long, Yiming Xu, Guang Yang, Shandian Zhe, Houman Owhadi
ICML 2025 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | In numerical experiments, we demonstrate the advantages of our method in solving several benchmark PDEs. Our implementation is released at https: //github.com/Bayesian AIGroup/ Efficient-Kernel-PDE-Solver. Experiments: Evaluation on Burgers , nonlinear elliptical, Eikonal, and Allen-Cahn equations have demonstrated our method s practical efficacy. |
| Researcher Affiliation | Academia | 1University of Utah 2University of Kentucky 3California Institute of Technology. Correspondence to: Shandian Zhe <EMAIL>. |
| Pseudocode | No | The paper describes its methodology in mathematical formulations and text, but does not include any explicitly labeled pseudocode or algorithm blocks. |
| Open Source Code | Yes | Our implementation is released at https: //github.com/Bayesian AIGroup/ Efficient-Kernel-PDE-Solver. |
| Open Datasets | No | The paper uses well-known benchmark PDEs (Burgers, nonlinear elliptical, Eikonal, Allen-Cahn equations). For these, solutions are either crafted (e.g., u(x) = sin(πx1) sin(πx2) + 4 sin(4πx1) sin(4πx2)) or computed via methods from other papers (e.g., Cole Hopf transformation with numerical quadrature or highly-resolved finite difference solver from Chen et al., 2021). No explicit publicly available datasets with direct access links or formal citations are provided. |
| Dataset Splits | No | The paper describes varying the number of collocation points (e.g., {600, 1200, 2400, 4800}) and using regularly-spaced square grids or specific spatial-to-time resolution ratios (e.g., 3:1). However, it does not specify traditional train/test/validation splits as understood in standard machine learning dataset evaluations; the collocation points are used to learn the PDE solution itself, not to partition a pre-existing dataset. |
| Hardware Specification | Yes | We ran the experiment on a Linux workstation equipped with an Intel(R) Xeon(R) Platinum 8360H Processor with 24GB memory. |
| Software Dependencies | No | We implemented our method with JAX (Frostig et al., 2018). The PINN is implemented with PyTorch (Paszke et al., 2019). Specific version numbers for JAX and PyTorch are not provided. |
| Experiment Setup | Yes | For SKS, we minimize (7) (with ϵ = 0), and used ADAM optimization with learning rate 10 3. The maximum number of epochs was set to 1M. We stopped the optimization if the loss stopped improving for 1K updates. For DAKS, we used the relaxed Gauss Newton optimization developed in the original paper. The PINN was first trained by 10K ADAM epochs with learning rate 10 3 and then by L-BFGS with learning rate 10 1 with a maximum of 50K iterations. The tolerance level for L-BFGS was set to 10 9. To identify the architecture for the PINN, we varied the number of layers from {2, 3, 5, 8, 10}, and the width of each layer from {10, 20, 30, . . . , 100}. We used tanh as the activation function. For DAKS and SKS, we used Square Exponential (SE) kernel with different length-scales across the input dimensions. We selected the nugget term from {5E-5, 1E-5, 5E-6, 1E-6, . . ., 1E-13}. For SKS, we selected α and β in (7) from the range {10 2, 10 1, . . . , 1010, 1012, 1014, 1015, 1020}, jointly with other hyperparameters. |