**Full Title: **DeepM&Mnet: A General Framework for Building Multiphysics & Multiscale Models using Neural Network Approximation of Functions, Functionals, and Nonlinear Operators (with JHU) sponsored by DARPA Computable Models Program.

**Abstract:** The machine learning community has made tremendous strides in the past 15 years by capitalizing on the neural network (NN) universal function approximation, and building a plethora of innovative networks with good generalization properties for diverse applications. However, it has ignored an even more powerful theoretical result thanks to Chen & Chen that

neural networks can in fact approximate functionals and even nonlinear operators with arbitrarily good accuracy! This is an astonishing result with significant implications, especially for modeling and simulation of physical systems, requiring accurate regression and not only approximate classification tasks as in commercial applications. We have produced preliminary results that document the potential breakthroughs in modeling complex engineering problems and different operators. For example, we developed DeepFnet that represents a functional predicting the dynamic motions of a destroyer battleship in extreme sea states, making predictions at a fraction of a second, unlike the OpenFoam CFD solver that takes one week per simulation. Similarly, we have developed DeepOnet to approximate integrals, ODEs, PDEs, and even fractional Laplacians by designing a new trunk-branch NN that approximates linear and nonlinear operators, and generalizes well to unseen functions.

Traditional methods, especially high-order discretizations such as WENO and spectral elements, can produce very accurate solutions of multiphysics and multiscale (M&M) problems but they do not scale well in high dimensions and large domains. Moreover, they cannot be easily combined with data and are prohibitively expensive for inverse problems. Real-world M&M problems are typically ill-posed with missing initial or boundary conditions and often only partially known physics, e.g. reactive transport. The Physics-Informed Neural Networks (PINNs) we have pioneered can tackle such problems given some extra (small) data anywhere in the domain. PINNs are easy to implement for multiphysics problems and particularly effective for inverse problems [10] but not as efficient or accurate for forward multiscale problems. Here, we propose a domain decomposition version of PINNs with conservative properties (CPINNs) resembling discontinuous Galerkin methods that can tackle multiscale problems effectively. Moreover, we propose new NNs (DeepFnet and DeepOnet) to approximate functionals and nonlinear operators as building blocks of a more general M&M development framework that can be used across disciplines for modeling M&M problems. We refer to this integrated framework as DeepM&Mnet, which can be used for any M&M problem in physics and engineering.

We plan to demonstrate CPINNs and DeepM&Mnet for diverse coupled PDE problems involving both smooth and discontinuous solutions. In particular, we will focus on high-speed aerodynamic flows in the hypersonic regime that involve 3 to 5 different fields depending on the Mach number as well as a wide range of temporal and spatial scales due to the inherent scale disparities in shock and turbulence phenomena. In the first phase, we will employ two diverse benchmark M&M problems to demonstrate the effectiveness of CPINNs and DeepM&Mnet, while in phase 2 we will simulate hypersonic transitional boundary layers (HTBLs) with finite-rate chemistry. Our team could also tackle any of the problems proposed by the DARPA team but we believe that the HTBL problem is the proper and strictest testbed for the following reasons: (i) Laminar-to-turbulence transition is a chaotic phenomenon that is notoriously sensitive to environmental uncertainties and hence accuracy is paramount in its prediction; (ii) Multiple physics are strongly coupled including fluid dynamics with discontinuities/shocks, far-from equilibrium thermodynamics with finite-rate chemistry, conjugate heat transfer with the walls; (iii) The ultimate flow state is turbulent, which is chaotic and multiscale. Note that data are very scarce for hypersonic applications and the physics are not always fully known, which renders the interpretation of limited measurements both ill-conditioned and not necessarily unique. This problem can also be expanded in the future to incorporate additional physics such as ablation, particle impact and damage. Therefore, hypersonic flows, which have significant implications for national security, are a formidable M&M challenge not amenable to conventional approaches, and an excellent testbed to develop DeepM&Mnet.