Operator Learning with Neural Fields: Tackling PDEs on General Geometries
Authors: Louis Serrano, Lise Le Boudec, Armand Kassaï Koupaï, Thomas X Wang, Yuan Yin, Jean-Noël Vittaut, Patrick Gallinari
NeurIPS 2023 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | 4 Experiments To demonstrate the versatility of our model, we conducted experiments on three distinct tasks (Figure 1): (i) solving an initial value problem (Section 4.1), (ii) modeling the dynamics of a physical system (Section 4.2), and (iii) learning to infer the steady state of a system based on the domain geometry (Section 4.3) plus an associated design problem in Appendix D. Since each task corresponds to a different scenario, we utilized task-specific datasets and employed different baselines for each task. This approach was necessary because existing baselines typically focus on specific tasks and do not cover the full range of problems addressed in our study, unlike CORAL. We provide below an introduction to the datasets, evaluation protocols, and baselines for each task setting. All experiments were conducted on a single GPU: NVIDIA RTX A5000 with 25 Go. Code will be made available. |
| Researcher Affiliation | Collaboration | 1 Sorbonne Universit e, CNRS, ISIR, 75005 Paris, France 2 Sorbonne Universit e, CNRS, LIP6, 75005 Paris, France 3 Criteo AI Lab, Paris, France |
| Pseudocode | Yes | We provide the pseudo-code in Algorithms 1 and 2. We present the inference procedure in Algorithm 4. |
| Open Source Code | Yes | The code is available at https://anonymous.4open.science/r/coral-0348/. |
| Open Datasets | Yes | We use the datasets from Pfaff et al. (2021), and take the first and last frames of each trajectory as the input and output data for the initial value problem. We consider the 2D Navier-Stokes equation as presented in Li et al. (2021); Yin et al. (2022). We generate the data with the Dedalus software (Burns et al., 2020), following the setting described in Yin et al. (2022), where a symmetric phenomena can be seen for both northern and southern hemisphere. The initial zonal velocity u0 contains two non-null symmetric bands in the both hemispheres, which are parallel to the circles of latitude. At each latitude and longitude ϕ, θ [ π 2 ] [ π, π]: en exp 1 (ϕ ϕ0)(ϕ ϕ1) , 0 if ϕ (ϕ0, ϕ1), umax en exp 1 (ϕ+ϕ0)(ϕ+ϕ1) , 0 if ϕ ( ϕ1, ϕ0), (0, 0) otherwise. We use the datasets provided by Li et al. (2022a) and adopt the original authors train/test split for our experiments. |
| Dataset Splits | Yes | We use a train, validation, test split of 1000 / 100 / 100 samples for all the evaluations. |
| Hardware Specification | Yes | All experiments were conducted on a single GPU: NVIDIA RTX A5000 with 25 Go. |
| Software Dependencies | No | We implemented all experiments with Py Torch (Paszke et al., 2019). This mentions PyTorch but not a specific version number. Other software like SU2 or COMSOL are mentioned for dataset generation but also without versions. |
| Experiment Setup | Yes | We provide the list of hyperparameters used for the experiments on Cylinder and Airfoil in Table 4. Table 5 summarizes the hyperparameters used in our experiments for dynamics modeling on datasets Navier-Stokes and Shallow-Water (Table 2). |