## PhILMs:Physics-Informed Learning Machines

### PhILMs: Physics-Informed Learning Machines for Multiscale and Multiphysics Problems

**Full Title: **PhILMS: Physics-Informed Learning Machines for Multiscale and Multiphysics Problems sponsored by the Department of Energy/Pacific Northwest National Laboratory** **

**Abstract:** Many U.S. Department of Energy (DOE) mission-critical applications require engineering materials at the nanoscale, optimizing multiscale processes and systems at the macroscale, and even the discovery of new governing physico-chemical laws across scales. Underlying all of these applications are the common grand challenges of cascades-of-scales; multiscale understanding; and scalable synthesis and processing in materials, physics, and chemistry. These scientific drivers require a deeper, broader, and more integrated understanding of common multiscale phenomena and scaling cascades, especially those arising due to the "hidden physics" of interfaces, inhomogeneities, symmetry-breaking and other "singularities." The new center, PhILMs (pronounced "films"), is at the intersection of mathematics, physics, data, and deep learning (DL). We will focus on complex systems governed by cascades-of-scales, e.g., functional materials, subsurface reactive transport, combustion, ice sheets, and other DOE applications that require new multiscale capabilities. Cascades-of-scales involve more than two scales with long-range spatio-temporal interactions that often lack self-similarity and proper closure relations. To tackle such problems, we propose a synthesis of physics-informed and data-driven tools and approaches, including nonlocal operators, multifidelity data and information fusion, DL, meshfree approximation theory, uncertainty propagation, and stochasticity. Our main focus is on predictability and uncertainty quantification of multiscale physical phenomena—not the usual classification or regression typically associated with DL. The high-level goals of PhILMs include developing physics-informed learning machines by encoding conservation laws and prior physical knowledge into Bayesian DL networks and analyzing their mathematical properties.

We demonstrate the effectiveness of PhILMs in designing functional materials with tunable properties and extending PhILMs to other DOE-relevant multiscale problems, e.g., combustion, subsurface, and Earth systems, where all exhibit inhomogeneous scaling cascades. We establish probabilistic scientific computing as a new meta-discipline at the interface of computational mathematics, multifidelity data, information fusion, and DL.

Our integrated mathematical and computational activities can be classified into four research areas: (RAI) partial differential equation (PDE)-based modeling of macroscales, (RA-II) stochastic modeling of mesoscales, (RA-III) bridging methods to connect the scales, and (RA-IV) Bayesian DL approximations and algorithms to support RA-I–RA-III. Specific benchmark problems for all areas, as well as integrated problems based on the driver applications, will be defined to measure our progress. Integrating these areas will lead to the development of PhILMs, and its usefulness in modeling multiscale phenomena and discovering new governing laws from multifidelity data will be demonstrated in the diverse driver applications motivated by DOE problems.

Our research team (based on the core of CM4) consists of two DOE laboratories and four universities. The adaptability of mathematical research to directions set by the DOE collaborators is of central importance. We will hold weekly webinars, annual workshops, and tutorials. We will form a committee to establish collaborative projects with stakeholders across the DOE laboratories. We will establish regular exchange visits for postdocs and students (when possible) to share their time between the universities and labs. We also will appoint an Advisory Committee to assist the project director and project leads.

PhILMs will forge a new direction focusing on the hidden physics of interfaces, inhomogeneous cascades and multiscale modeling, based on new developments in Bayesian DL, nonlocal operators, and concurrent coupling. PhILMs will promote rapid information dissemination to DOE researchers and effective training for a new cadre of modern multiscale/data modeling scientists.