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Full Title: Learning and Meta-Learning of Partial Differential Equations via Physics-Informed Neural Networks: Theory, Algorithms, and Applications sponsored by Air Force Office of Scientific Research Multidisciplinary Research Program of the University Research Initiative (AFOSR MURI)

Abstract: Research Problem: Despite the significant progress over the last 50 years in simulating multiphysics problems using numerical discretization of partial differential equations (PDEs), we still cannot incorporate seamlessly noisy data into existing algorithms, mesh-generation is complex, and we cannot tackle high-dimensional problems governed by parametrized PDEs. Moreover, solving inverse problems is often prohibitively expensive and requires different formulations and new computer codes. We propose to overcome these obstacles by introducing physics-informed learning, integrating seamlessly data and mathematical models, and implementing them using physics-informed neural networks (PINNs) and other new physics- informed networks (PINs). We will blend knowledge on existing methods, e.g. domain decomposition and uncertainty quantification, with the new concepts in deep neural networks and more general networks and regressions. Inversely, we will employ synergistically the lessons learned from PINNs/PINs to enhance the performance of existing numerical methods, e.g., for low-dimensional modeling and high-dimensional PDEs.

Technical Approach: We will develop new theory and algorithms for learning and meta- learning (ML) of physical systems via PINNs and PINs, which are interpretable, generalizable and lead to reproducible results. ML will be the basis of generalization and transfer learning by efficiently searching for novel architectures tailored to the data and the physical problem modeled. We will primarily focus on inverse problems, possibly-ill posed like in non-destructive evaluation (NDE) of materials, as well as aerodynamic and material design of cruise and accelerating hypersonic vehicles involving thermo-fluid-structure interactions governed by cascades-of-scales and subject to large uncertainties. To tackle such problems, we propose a synthesis of classical and new approaches, including domain-decomposition and adaptivity, multifidelity data and information fusion, optimization, adversarial training, a new Bayesian framework for analysis of PINNs and PINs, and a new point of view based on the Hamilton-Jacobi PDE. The proposed research plan emphasizes a dynamic integration of six main research areas that form the pillars of our comprehensive integrated approach.

Outcome: Mathematical theory of PINNs/PINs; Faster optimization algorithms; Uncertainty quantification for PINNs; Meta-learning for PDEs; General game-theoretic/adversarial framework; Parallel PINNs; Efficient model reduction in high-dimensions; Solutions of stochastic PDEs in high-dimensions; Hamilton-Jacobi based theory related to deep learning; PINN- based design framework for hypersonic vehicles; PINN-based NDE of materials (algorithms and code) delivered to Air Force; Benchmark examples and open source codes on Github; Outreach and dissemination program.

Impact: We propose a paradigm shift in computational science and engineering for physics- informed learning to solve problems with partially known physics, which cannot be easily solved with existing methods. This new generation of PINNs/PINs will be optimized automatically via meta-learning. We will train a new cadre of simulation scientists in the classroom but also by reaching out and collaborating with researchers at different DoD labs. The proposed work is to develop technology for both military and civil applications.