Learning Neural PDE Solvers with Convergence Guarantees
Authors: Jun-Ting Hsieh, Shengjia Zhao, Stephan Eismann, Lucia Mirabella, Stefano Ermon
ICLR 2019 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | After training on a single geometry, our model generalizes to a wide variety of geometries and boundary conditions, and achieves 2-3 times speedup compared to state-of-the-art solvers. We evaluate our method on the 2D Poisson equation with Dirichlet boundary conditions, 2u = f. We train our model on simple domains where the ground truth solutions can be easily obtained, and then evaluate its performance on different geometries and boundary conditions. Table 1 shows results of the Conv model. Figure 2 shows that our model is comparable or faster than FEni CS in wall clock time. |
| Researcher Affiliation | Collaboration | Jun-Ting Hsieh* Stanford junting@stanford.edu Shengjia Zhao* Stanford sjzhao@stanford.edu Stephan Eismann Stanford seismann@stanford.edu Lucia Mirabella Siemens lucia.mirabella@siemens.com Stefano Ermon Stanford ermon@stanford.edu |
| Pseudocode | No | The paper describes mathematical formulations and updates (e.g., Eq. 5, 6, 7, 8, 9, 10) but does not include any blocks explicitly labeled as "Pseudocode" or "Algorithm". |
| Open Source Code | No | The paper does not contain any statement about releasing open-source code for the described methodology, nor does it provide a link to a code repository. |
| Open Datasets | No | To reemphasize, our goal is to train a model on simple domains where the ground truth solutions can be easily obtained, and then evaluate its performance on different geometries and boundary conditions. Therefore, for training, we select the simplest Laplace equation, 2u = 0, on a square domain with boundary conditions such that each side is a random fixed value. Figure 1a shows an example of our training domain and its ground truth solution. |
| Dataset Splits | No | The paper specifies training on a square domain and testing on various geometries and larger grid sizes, but it does not explicitly define training, validation, and test splits with percentages or sample counts. |
| Hardware Specification | Yes | On GPU, we measure an additional 30 speedup (on Tesla K80 GPU, compared with a 64-core CPU). |
| Software Dependencies | No | The paper mentions using the "FEni CS package (Logg et al., 2012)" but does not specify a version number for FEni CS or any other software dependencies. It also refers to "standard convolutional networks" but not specific libraries or their versions. |
| Experiment Setup | Yes | k in our experiments is uniformly chosen from [1, 20], similar to the procedure in (Song et al., 2017). For the FEni CS model, We set the error threshold to be 1 percent of the initial error. |