ABSTRACT

Many phenomena in solid and fluid mechanics are modeled via complex partial differential equations (PDEs). Since solving these PDEs via traditional numerical methods is prohibitively expensive, emulators such as deep neural networks (DNNs) are increasingly employed to approximate PDE solutions. While significant effort has been expended in this direction, existing technologies provide expensive solutions that are not transferable across different applications or scalable to complex PDEs. This project aims to address these limitations using a divide and conquer approach. In the framework that will be developed for the project, the project will first build a library of DNNs that solve single-physics PDE systems over small domains called genomes. Then, to solve multi-physics PDEs over large unseen domains, the project will develop an adaptive method that couples the DNNs and assembles their genome-wise predictions such that the governing equations are satisfied in the entire domain. The project expects that the pre-trained DNNs and coupling mechanism will greatly benefit scholars without access to the hardware or knowledge that are needed for scientific machine learning. The transferability of the framework has the potential to reduce the carbon footprint of the high computing costs that are associated with existing technologies that use DNNs to solve PDEs, providing great benefits to both scientific research and to society as a whole.

The project will build LEarned Genomic Operators (LEGOs) that use Bayesian reinforcement learning (BRL) for generalization, i.e., for (1) emulating multi-physics systems, and/or (2) achieving spatiotemporal transferability and scalability. The contributions of this work are expected to enable on-the-fly approximation of the behavior of solids and fluids via pre-trained DNNs, thus eliminating long training times while increasing accuracy and scalability. The LEGO framework is hypothesis-driven and leverages the mathematics of domain decomposition methods that uniquely exploit parallel and heterogeneous machines. The framework solves a PDE system in a large domain with arbitrary initial and boundary conditions by first decomposing the domain into small subdomains called genomes. Then, the solution in each genome is approximated via pre-trained LEGOs such that the assembly of the genome-wise predictions approximates the solution in the large domain. In essence, the LEGOs model different physical phenomena (e.g., material deformation or fluid flow) in genomes while the BRL agent couples the LEGOs to model multi-physics phenomena and/or spatiotemporally extends the predictions of LEGOs while preserving solution consistency across the genomes. To achieve real-time and robust performance with high transferability and scalability, the framework (1) uses mixed-precision computing and hardware accelerators, (2) incorporates geometry-aware learning algorithms, and (3) mathematically estimates the propagated errors during solution assembly.

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.

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