Notice: The reproducibility variables underlying each score are classified using an automated LLM-based pipeline, validated against a manually labeled dataset. LLM-based classification introduces uncertainty and potential bias; scores should be interpreted as estimates. Full accuracy metrics and methodology are described in Coakley et alK. L. Coakley, T. Snelleman, H. Hoos, and O. E. Gundersen, "The embrace of open science: An analysis of a decade of AI research and 56 800 conference papers," Under Review, 2026..
PINNs with Learnable Quadrature
Authors: Sourav Pal, Kamyar Azizzadenesheli, Vikas Singh
NeurIPS 2025 | Venue PDF | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | We demonstrate the effectiveness of our proposed framework involving the learnable quadrature module (Learn Quad) in solving PDEs via PINNs next. First we compare the empirical performance in solving single PDEs which demonstrate their effectiveness when used with PINNs. We emphasize that the use of Learn Quad improves the performance of PINNs and helps close the gap with numerical methods based solvers. Thereafter, we describe how the use of hyper-networks can enable Learn Quad to efficiently solve multiple instances of a given PDE. Our extensive set of baselines include: PINN(fixed), PINN(dynamic) [Karniadakis et al., 2021], Curr Reg [Krishnapriyan et al., 2021], CPINN (fixed, dynamic) Wang et al. [2022], RAR-G [Lu et al., 2021], RAD [Nabian et al., 2021], RAR-D [Wu et al., 2023], R3 and Causal R3[Daw et al., 2023]. We performed experiments on benchmark PDEs considered in the baselines. This ensured that we are able to clearly demonstrate the advantage of Learn Quad. |
| Researcher Affiliation | Collaboration | Sourav Pal UW-Madison EMAIL Kamyar Azizzadenesheli NVIDIA Corporation EMAIL Vikas Singh UW-Madison EMAIL |
| Pseudocode | Yes | Algorithm 1 Training for a single PDE Algorithm 2 Training for a family of PDEs |
| Open Source Code | Yes | Our code is available at https://github.com/vsinghgroup/learn-quad. |
| Open Datasets | No | The paper generates instances of PDEs using parametric forms and distributions for forcing functions and boundary/initial conditions, e.g., sampling parameters from a uniform distribution U(1, 2.5). This is not using pre-existing publicly available datasets, but rather generating problem instances for experimentation. Thus, no specific open datasets are provided. |
| Dataset Splits | Yes | For each family, we sampled 100 instances of the PDE and used a train/test split of 80/20. |
| Hardware Specification | Yes | All experiments were performed on a single NVIDIA 2080 Ti GPU. |
| Software Dependencies | No | The paper mentions using 'JAX' and 'Pytorch' (for baselines) but does not provide specific version numbers for any software dependencies. |
| Experiment Setup | Yes | The number of parameters used for the learnable weight function in the Learn Quad module was roughly 500 parameters in all cases. All neural networks were implemented using fully connected layers with tanh as the activation function. All experiments were performed on a single NVIDIA 2080 Ti GPU. The number of epochs used for diffusion PDE was 100k while for Burger s, Wave and Allen-Cahn PDE they were run for 200k epochs. This was determined empirically based on convergence of the L2 relative error. We used a learning rate of 1e-3. |