Classical time-stepping for parabolic PDEs is stable, but it becomes expensive on fine grids. Networks trained only on a residual in the continuum — physics-informed neural networks (PINNs) — often struggle with discrete structure and boundary conditions. I work on numerics-informed solvers: the network is trained on a discrete residual of a chosen scheme (in the thesis, Backward Euler with a five-point Laplacian), with exact Dirichlet conditions enforced by boundary lifting. At Sandia I reused those pretrained local solvers as subdomain problems inside overlapping Schwarz domain decomposition.

Projects · Earlier work · Publications · Talks

Research projects

Numerics-informed neural networks for parabolic PDEs

Rice University · advisor Dr. Beatrice Riviere

I extend Numerics-Informed Neural Networks (NINNs) for two-dimensional parabolic problems, including the heat equation. NINNs were introduced by Celaya, Kirk, Fuentes, and Riviere (2024) (arXiv:2311.00259; code aecelaya/pde-nets) as unsupervised, finite-difference–based convolutional solvers.

A compact U-Net is trained on the discrete residual of Backward Euler plus a five-point Laplacian. Dirichlet data enter through boundary lifting rather than penalty terms. I compare this solver to PINNs and to the same finite-difference scheme.

I implemented the PyTorch training pipelines, ran GPU experiments on Rice NOTS, and analyzed manufactured-solution error, stability, and convergence across training schedules. On selected smooth problems the NINN shows at least second-order convergence and smaller error than the same finite-difference scheme.

The work is written up in the M.A. thesis (May 2026). A journal version with Adrian Celaya and Beatrice Riviere is in preparation.

Hybrid NINN–FOM overlapping Schwarz

Sandia CSRI · Summer 2026 · mentored by Dr. Irina Tezaur

Overlapping Schwarz iterates on overlapping subdomains coupled through interface data. I built a Python/PyTorch codebase for two-dimensional advection–diffusion. Each subdomain can be a classical finite-difference full-order model (FOM) or a NINN.

I trained NINNs offline on discrete residual losses with exact Dirichlet conditions and reused them as local solvers instead of retraining every Schwarz iteration. I compared hybrid couplings to all-classical Schwarz at high Péclet number on held-out cases, and I extended the same coupling to time-dependent problems over successive time windows.

This coupling is arXiv:2609.17841 (September 2026), with Irina Tezaur, Alejandro Diaz, and Beatrice Riviere. It builds on Celaya et al.’s NINNs and on Schwarz–PINN coupling by Snyder, Tezaur, and Wentland (2023).

HiOp GPU/RAJA driver

Lawrence Livermore National Laboratory · Summer 2025

I worked in HiOp (a high-performance optimization library), using RAJA (a portability layer for parallel loops on CPUs and GPUs).

I implemented a RAJA-based dense-constraint driver with MPI in HiOp, so the same code path can run on CPU and NVIDIA GPU backends. I added GPU memory-space options and MAGMA GPU paths for limited-memory quasi-Newton components, alongside existing LAPACK CPU paths, plus device-agnostic vector and matrix kernels with RAJA parallel loops and unified memory.

I configured GPU builds and tests on LLNL’s Lassen supercomputer (IBM Power9 + NVIDIA V100) with CMake, automated ctest, and jsrun, and I debugged GPU code with TotalView. The workflow used Git feature branching, code reviews, and Umpire.

Earlier work

Before focusing on neural PDE solvers, I studied discontinuous Galerkin formulations for coupled flow and deformation in porous media, and I explored phase-field models of fracture.

During my M.S. at San José State University, advised by Dr. Slobodan Simić, I applied dynamical systems methods to cosmological models in general relativity, including stability analysis of the ΛCDM model.

Publications and theses

• Hybrid coupling with numerics-informed neural networks and the overlapping Schwarz alternating method.
  George Chumbipuma, Irina Tezaur, Alejandro Diaz, and Beatrice Riviere. arXiv:2609.17841 [cs.LG], September 2026.

• Numerics-Informed Neural Networks for Parabolic Partial Differential Equations.
  Jorge Chumbipuma. M.A. thesis, Department of Computational Applied Mathematics and Operations Research, Rice University, May 2026. Advisor: Beatrice Riviere.