Advanced Neural Network-Enhanced Reduced Order Modeling for Solving PDEs with High-Dimensionality, Randomness, and Multiscale Features

Many challenges in science and engineering require fast and reliable predictions of how physical systems change over time, such as heat moving through a room, gas spreading through air, fluid flowing through complex materials, or environmental conditions changing around an autonomous robot. These predictions are often made by solving partial differential equations, but the calculations can become too slow or expensive when the system includes many variables, uncertain inputs, changing conditions, or small-scale details that strongly affect the overall behavior. This project will develop new mathematical and artificial intelligence methods that make these simulations faster while keeping them accurate and consistent with governing physical principles. The work can strengthen the foundations of reliable scientific computing and help scientists and engineers test more scenarios, update predictions as conditions change, and make better decisions in time-sensitive settings. Potential benefits include improved environmental prediction, engineering design, scientific modeling, and safer autonomous robotic systems that respond to changing heat, airflow, gas, or transport conditions. The project will also support education and workforce development by training graduate and undergraduate students in computational mathematics, numerical methods, machine learning, and data science; incorporating research outcomes into courses and seminars; supporting summer research experiences; and engaging local students through mathematics outreach. This project will develop neural network-enhanced reduced order modeling methods for partial differential equations with high dimensionality, randomness, nonlinear behavior, and multiscale features. The research will combine the stability and efficiency of reduced order models with the flexibility of neural networks. One research direction will develop hybrid operator-learning reduced models in which precomputed multiscale basis functions capture known physical structures, while neural network components generate additional basis functions that adapt to changing parameters, random inputs, geometries, and solution features. Stability-preserving Petrov-Galerkin formulations, operator-informed test spaces, and residual-based error indicators will guide training and basis enrichment. A second direction will develop transfer learning methods for families of related partial differential equations. Instead of retraining a full neural network for every new equation, the method will learn reusable feature representations across related operators and adapt them through lightweight updates when coefficients, source terms, boundary conditions, geometries, or random inputs change. Similarity measures based on operator information, spectral behavior, and learned solution features will guide decisions about when transfer is appropriate and when additional adaptation is needed. A third direction will combine these stable reduced representations and transferable neural features to build fast, adaptive surrogate models for high-dimensional and multiscale problems. The methods will be evaluated using benchmark partial differential equations and environmental field prediction problems for autonomous robotics, including temperature, gas concentration, airflow-driven transport, and related diffusion or advection-diffusion processes. Expected contributions include new stability-aware neural reduced models, new transfer learning tools for nonlinear partial differential equation families, and efficient simulation methods for scientific computing and engineering applications. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.