Pseudo-Physics-Informed Neural Operators: Enhancing Operator Learning from Limited Data
Authors: Keyan Chen, Yile Li, Da Long, Zhitong Xu, WEI W. XING, Jacob Hochhalter, Shandian Zhe
TMLR 2025 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | The effectiveness of PPI-NO is validated through extensive evaluations on five commonly used benchmark operator learning tasks in literature (Li et al., 2020b; Lu et al., 2022), including Darcy flow, nonlinear diffusion, Eikonal, Poisson and advection equations, as well as one application in fatigue modeling in fracture mechanics, where the ground-truth holistic PDE system is unknown. Our implementation is released at https://github.com/BayesianAIGroup/PPI_NO |
| Researcher Affiliation | Academia | Keyan Chen* EMAIL Kahlert School of Computing University of Utah Yile Li* EMAIL Kahlert School of Computing University of Utah Da Long EMAIL Kahlert School of Computing University of Utah Zhitong Xu EMAIL Kahlert School of Computing University of Utah Wei W. Xing EMAIL School of Mathematical and Physical Sciences University of Sheffield Jacob Hochhalter EMAIL Department of Mechanical Engineering University of Utah Shandian Zhe EMAIL Kahlert School of Computing University of Utah |
| Pseudocode | Yes | Algorithm 1 Pseudo-Physics-Informed NO |
| Open Source Code | Yes | Our implementation is released at https://github.com/BayesianAIGroup/PPI_NO |
| Open Datasets | Yes | The effectiveness of PPI-NO is validated through extensive evaluations on five commonly used benchmark operator learning tasks in literature (Li et al., 2020b; Lu et al., 2022), including Darcy flow, nonlinear diffusion, Eikonal, Poisson and advection equations, as well as one application in fatigue modeling in fracture mechanics, where the ground-truth holistic PDE system is unknown. The details about all the datasets are given in Section A of the Appendix. |
| Dataset Splits | Yes | For each operator learning benchmark, we simulated 100 examples for testing, and varied the number of training examples from {5, 10, 20, 30}, except for Advection, we ran with {20, 30, 50, 80} training examples. For SIF prediction, which is much more challenging, we experimented with training size from {400, 500, 600}, and employed 200 test examples. We repeated the evaluation for five times, each time we randomly sampled a different training set. |
| Hardware Specification | Yes | all measurements are obtained on a Linux workstation equipped with an NVIDIA GeForce RTX 4090 (24 GB). |
| Software Dependencies | No | All the models were implemented by PyTorch (Paszke et al., 2019), and optimized with ADAM (Kingma & Ba, 2014). |
| Experiment Setup | Yes | For the pseudo physics neural network ϕ see equation 3 we tuned the kernel size from {(3, 3), (5, 5), (7, 7), (9, 9)}. The stride was set to 1 and padding was set to same to ensure the output shape does not change. In the subsequent fully connected layers, we chose the number of layers from {3, 4, 5, 6}, and the layer width from {16, 32, 64}. We used GeLU activation. For FNO, we set the number of modes to 12 and channels to 32 (in the lifted space). We varied the number of Fourier layers from {2, 3, 4}. For DONet, in all the cases except Darcy Flow, the trunk net and branch net were constructed as fully connected layers. We varied the number of layers from {2, 3, 4} and the layer width was chosen from {30, 40, 50, 60}, with ReLU activation. For the case of Darcy flow, we found that DONet with only fully connected layers exhibited inferior performance. To address this, we introduced convolution layers into the branch net. We selected the number of convolution layers from {3,5,7}, and employed batch normalization and leaky ReLU after each convolution layer. To incorporate the learned pseudo physics representation into the training of FNO or DONet, we randomly sampled 200 input functions to construct the second loss term in equation 5. We set the maximum number of iterations to 10 and selected the weight λ from [10^-1, 10^2]. All the models were implemented by PyTorch (Paszke et al., 2019), and optimized with ADAM (Kingma & Ba, 2014). The learning rate was selected from {10^-4, 5x10^-4, 10^-3}. The number of epochs for training or fine-tuning FNO, DONet and pseudo physics network ϕ was set to 500 to ensure convergence. |