Physics informed neural networks (PINNs) have emerged as a powerful tool to provide robust and accurate approximations of solutions to partial differential equations (PDEs). However, PINNs face serious difficulties and challenges when trying to approximate PDEs with dominant hyperbolic character. This research focuses on the development of a physics informed deep learning framework to approximate solutions to nonlinear PDEs that can develop shocks or discontinuities without any a-priori knowledge of the solution or the location of the discontinuities. The work takes motivation from finite element method that solves for solution values at nodes in the discretized domain and use these nodal values to obtain a globally defined solution field. Built on the rigorous mathematical foundations of the discontinuous Galerkin method, the framework naturally handles imposition of boundary conditions (Neumann/Dirichlet), entropy conditions, and regularity requirements. Several numerical experiments and validation with analytical solutions demonstrate the accuracy, robustness, and effectiveness of the proposed framework.

This work presents a physics-informed deep learning-based super-resolution framework to enhance the spatio-temporal resolution of the solution of time-dependent partial differential equations (PDE). Prior works on deep learning-based super-resolution models have shown promise in accelerating engineering design by reducing the computational expense of traditional numerical schemes. However, these models heavily rely on the availability of high-resolution (HR) labeled data needed during training. In this work, we propose a physics-informed deep learning-based framework to enhance the spatial and temporal resolution of coarse-scale (both in space and time) PDE solutions without requiring any HR data. The framework consists of two trainable modules independently super-resolving the PDE solution, first in spatial and then in temporal direction. The physics based losses are implemented in a novel way to ensure tight coupling between the spatio-temporally refined outputs at different times and improve framework accuracy. We analyze the capability of the developed framework by investigating its performance on an elastodynamics problem. It is observed that the proposed framework can successfully super-resolve (both in space and time) the low-resolution PDE solutions while satisfying physics-based constraints and yielding high accuracy. Furthermore, the analysis and obtained speed-up show that the proposed framework is well-suited for integration with traditional numerical methods to reduce computational complexity during engineering design.

This work presents a novel physics-informed deep learning based super-resolution framework to reconstruct high-resolution deformation fields from low-resolution counterparts, obtained from coarse mesh simulations or experiments. We leverage the governing equations and boundary conditions of the physical system to train the model without using any high-resolution labeled data. The proposed approach is applied to obtain the super-resolved deformation fields from the low-resolution stress and displacement fields obtained by running simulations on a coarse mesh for a body undergoing linear elastic deformation. We demonstrate that the super-resolved fields match the accuracy of an advanced numerical solver running at 400 times the coarse mesh resolution, while simultaneously satisfying the governing laws. A brief evaluation study comparing the performance of two deep learning based super-resolution architectures is also presented.

Numerical methods such as Finite element method [Hug12], Isogeomteric analysis [CHB09], and mesh-free methods [LJZ95, BLG94] are few of the conventional approaches employed in solving the Partial Differential Equations (PDEs) involved in computational solid mechanics problems. However, the ever-increasing sophistication of material models by incorporating more complex physics, such as modeling size-effect [FMAH94, AA20b] or dislocation density evolution [AZA20, Aro19, AA20a, AAA22, JABG20], or advanced materials such as composites and multicomponent alloys with spatially-varying material properties (heterogeneity) and direction dependent behavior (anisotropy) is bringing these numerical solvers to their limits. Hence, it is becoming a formidable task to perform simulations that can resolve the complex physical phenomena occurring at small spatial and temporal scales and accurately predict the macro-scale behavior of materials. Therefore, a cost-effective physicsbased surrogate model that allows the researchers to perform simulations on a coarse mesh without sacrificing accuracy will be highly beneficial for many reasons. First, researchers can choose to run their simulations at a lower resolution (online stage) and later reconstruct the solution back to the target resolution (offline stage). This will significantly reduce the computational expense during the online stage, thus accelerating the process of scientific investigation and discovery. Second, the surrogate model based on data super-resolution can also be used to enhance outputs from experimental techniques for full-field displacement and strain measurement such as Digital Image Correlation (DIC) which would allow researchers to generate and store a small fraction of data. Recent advances in Deep Learning (DL) and Physics-Informed Neural Networks (PINN) [RPK17, RPK19] make it a promising tool to tackle this computational challenge.

This work presents a physics-informed neural network (PINN) based framework to model the strain-rate and temperature dependence of the deformation fields in elastic-viscoplastic solids. To avoid unbalanced back-propagated gradients during training, the proposed framework uses a simple strategy with no added computational complexity for selecting scalar weights that balance the interplay between different terms in the physics-based loss function. In addition, we highlight a fundamental challenge involving the selection of appropriate model outputs so that the mechanical problem can be faithfully solved using a PINN-based approach. We demonstrate the effectiveness of this approach by studying two test problems modeling the elastic-viscoplastic deformation in solids at different strain rates and temperatures, respectively. Our results show that the proposed PINN-based approach can accurately predict the spatio-temporal evolution of deformation in elastic-viscoplastic materials.