Numerics-Informed Neural Networks (NINNs) for PDEs

Under the direction of Beatrice Riviere, I study and extend Numerics-Informed Neural Networks (NINNs) for parabolic PDEs. NINNs were introduced by Celaya, Kirk, Fuentes, and Riviere (2024) (arXiv:2311.00259; code aecelaya/pde-nets) as unsupervised, finite-difference–based convolutional solvers. My work adapts and evaluates that methodology—rather than inventing NINNs or PINNs—through PyTorch implementations, training-schedule studies, and HPC experiments.

Contributions:

• Adapted Celaya-style Backward Euler NINNs with boundary lifting (versus penalty BCs) and compact U-Net choices for two-dimensional parabolic problems
• Implemented PyTorch training pipelines and GPU-aware experiment campaigns (including on Rice NOTs), with ablations and reproducible artifacts in neural-pde-solvers
• Analyzed error propagation, stability, and convergence across training schedules and test problems, comparing NINNs to PINNs and classical finite-difference time-stepping
• Documented the computational study in an MA thesis (December 2025); a journal manuscript with Adrian Celaya and Beatrice Riviere is in preparation

Hybrid Neural / Full-Order PDE Coupling (Sandia Summer 2026)

During a summer internship at Sandia National Laboratories, mentored by Irina Tezaur and collaborators, I built pde-solver-lab—a greenfield research lab for classical finite differences, overlapping Schwarz iteration, FD-NINNs, PINN baselines, and hybrid NINN–FOM studies. The science directions and coupling ideas build on prior work, including Celaya et al.’s NINNs and Schwarz–PINN coupling by Snyder, Tezaur, and Wentland (2023).

Contributions:

• Designed and implemented the experiment stack (solvers, coupling loops, notebooks, tests, and campaign tooling) used for summer studies
• Reimplemented and validated Celaya-style NINNs in PyTorch as reusable subdomain solvers within hybrid settings
• Ran controlled studies of hybrid NINN–FOM (and related) overlapping Schwarz couplings, including boundary-condition and training/reuse choices directed by mentors

Accurate framing: I engineered the software and executed the campaigns. I did not invent NINNs, PocketNet, or Schwarz–PINN coupling.

High-Performance Optimization (LLNL Summer 2025)

During my summer internship at Lawrence Livermore National Laboratory, I worked on GPU-enabled optimization in the HiOp (High-performance Optimization) framework.

Contributions:

• Implemented a RAJA-based nonlinear dense constraint driver and solver with MPI support, enabling portable performance across CPU and NVIDIA GPU backends
• Ported limited-memory quasi-Newton (QN) methods to GPU architectures by threading a memory-space option throughout solver components and replacing CPU-only LAPACK calls with GPU-ready MAGMA and cuSOLVER placeholders
• Refactored HiOp's linear algebra layer to introduce device-agnostic kernels, RAJA parallel loops, and unified memory (UM) support for efficient host–device data movement
• Designed and documented GPU build/test workflows on LLNL's Lassen supercomputer (IBM Power9 + NVIDIA V100), including automated ctest parallel testing and jsrun-based job launches
• Resolved GPU-related issues using TotalView, cuda-memcheck, and RAJA execution policies
• Followed LLNL development practices including Git feature branching, pull requests, code reviews, and Umpire-aware memory management

Earlier Explorations (2023–2024)

Prior to focusing on neural PDE solvers, I explored several numerical methods and applications:

• Previously studied discontinuous Galerkin formulations for coupled flow and deformation in porous media.
• Explored phase-field approaches for fracture modeling as part of early numerical method studies.

These explorations continue to inform my perspective on multiscale and multiphysics simulation challenges.

Dynamical Systems and Cosmology (SJSU Master's Research)

During my master’s studies at San José State University, my research focused on applying dynamical systems theory to cosmological models in general relativity. This work provided a mathematical framework to study the evolution of the universe and analyze the stability of its critical points.

Focus:

• Lambda Cold Dark Matter (ΛCDM) Model: Analyzed the stability of critical points in the ΛCDM model, examining transitions between radiation-dominated, matter-dominated, and dark energy-dominated phases of the universe.
• Geometric Insights: Used dynamical systems techniques to explore the relationship between geometry and energy in cosmological equations.
• Numerical Simulations: Conducted simulations to verify theoretical findings and visualize trajectories of the universe’s evolution.

Outcomes:

• Improved understanding of the long-term behavior of cosmological systems in the models studied.
• Explored dynamical-systems techniques for analyzing nonlinear cosmological equations.

This research examined the interplay between mathematics and physics, and continues to inform my work on complex systems in applied mathematics.

Publications & Theses

• Master’s thesis. Jorge Chumbipuma. Numerics-Informed Neural Networks for Parabolic Partial Differential Equations. M.A. thesis, Department of Computational Applied Mathematics and Operations Research, Rice University, December 2025. Advisor: Beatrice Riviere.
• Manuscript in preparation. Jorge Chumbipuma, Adrian Celaya, and Beatrice Riviere. Numerics-informed neural networks for parabolic equations (working title).
• Related foundational work (not my authorship). Adrian Celaya, Keegan Kirk, David Fuentes, and Beatrice Riviere. Solutions to elliptic and parabolic problems via finite difference based unsupervised small linear convolutional neural networks. Computers & Mathematics with Applications, 174:31–42, 2024. Also arXiv:2311.00259. Introduces the NINN methodology that my thesis and ongoing work extend.