Physics-guided Neural Networks (PGNNs) represent an emerging class of neural networks that are trained using physics-guided (PG) loss functions (capturing violations in network outputs with known physics), along with the supervision contained in data. Existing work in PGNNs has demonstrated the efficacy of adding single PG loss functions in the neural network objectives, using constant tradeoff parameters, to ensure better generalizability. However, in the presence of multiple PG functions with competing gradient directions, there is a need to adaptively tune the contribution of different PG loss functions during the course of training to arrive at generalizable solutions. We demonstrate the presence of competing PG losses in the generic neural network problem of solving for the lowest (or highest) eigenvector of a physics-based eigenvalue equation, which is commonly encountered in many scientific problems. We present a novel approach to handle competing PG losses and demonstrate its efficacy in learning generalizable solutions in two motivating applications of quantum mechanics and electromagnetic propagation. All the code and data used in this work are available at https://github.com/jayroxis/Cophy-PGNN.

Generating flow fields (such as pressure and velocity fields) in 3D space is a fundamental task in computational fluid dynamics (CFD), with applications across a vast spectrum of science and engineering problems. An important class of fluid flow problems in CFD is multi-phase flow, where dispersed solid particles are present in the fluid flow. Despite recent developments in deep learning (DL) for CFD applications, current state-of-the-art is still unable to model 3D flow fields, especially in multi-phase flow settings. It is with this goal that we introduce PhyFlow, a novel physics-guided deep learning architecture for modeling 3D multi-phase fluid flows, designed to mimic the popular projection method for solving fluid flows in CFD simulations. We demonstrate that PhyFlow generates high quality flow fields and yields a 49.61% improvement over other state-of-the-art baselines. We also test the quality of PhyFlow based fields by employing them in downstream tasks like particle drag force prediction and demonstrate state-of-the-art results, improving upon the previous best models by 9.89%. Finally, we demonstrate the consistency of PhyFlow predictions with known underlying physics governing equations. Our source code and data are available online*.*tinyurl.com/mjkcrsdw

We propose quadratic residual networks (QRes) as a new type of parameter-efficient neural network architecture, by adding a quadratic residual term to the weighted sum of inputs before applying activation functions. With sufficiently high functional capacity (or expressive power), we show that it is especially powerful for solving forward and inverse physics problems involving partial differential equations (PDEs). Using tools from algebraic geometry, we theoretically demonstrate that, in contrast to plain neural networks, QRes shows better parameter efficiency in terms of network width and depth thanks to higher non-linearity in every neuron. Finally, we empirically show that QRes shows faster convergence speed in terms of number of training epochs especially in learning complex patterns.

Physics-based simulations are often used to model and understand complex physical systems in domains like fluid dynamics. Such simulations although used frequently, often suffer from inaccurate or incomplete representations either due to their high computational costs or due to lack of complete physical knowledge of the system. In such situations, it is useful to employ machine learning to fill the gap by learning a model of the complex physical process directly from simulation data. However, as data generation through simulations is costly, we need to develop models being cognizant of data paucity issues. In such scenarios it is helpful if the rich physical knowledge of the application domain is incorporated in the architectural design of machine learning models. We can also use information from physics-based simulations to guide the learning process using aggregate supervision to favorably constrain the learning process. In this paper, we propose PhyNet, a deep learning model using physics-guided structural priors and physics-guided aggregate supervision for modeling the drag forces acting on each particle in a Computational Fluid Dynamics-Discrete Element Method (CFD-DEM). We conduct extensive experiments in the context of drag force prediction and showcase the usefulness of including physics knowledge in our deep learning formulation. PhyNet has been compared to several state-of-the-art models and achieves a significant performance improvement of 8.46% on average. The source code has been made available∗ and the dataset used is detailed in [1, 2].