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Many engineering and scientific fields have recently become interested in modeling terms in partial differential equations (PDEs) with neural networks, which requires solving the inverse problem of learning neural network terms from…
We consider the primal and dual forms of the optimality conditions for PDE-contrained optimization problems arising in Data-Driven Computational Mechanics when specialized to the reaction-diffusion context. Starting with the continuous…
In this paper, we present a new framework how a PDE with constraints can be formulated into a sequence of PDEs with no constraints, whose solutions are convergent to the solution of the PDE with constraints. This framework is then used to…
A computed approximation of the solution operator to a system of partial differential equations (PDEs) is needed in various areas of science and engineering. Neural operators have been shown to be quite effective at predicting these…
Many practical applications require solving an optimization over large and high-dimensional data sets, which makes these problems hard to solve and prohibitively time consuming. In this paper, we propose a parallel distributed algorithm…
A design optimization framework for process parameters of additive manufacturing based on finite element simulation is proposed. The finite element method uses a coupled thermomechanical model developed for fused deposition modeling from…
The term `surrogate modeling' in computational science and engineering refers to the development of computationally efficient approximations for expensive simulations, such as those arising from numerical solution of partial differential…
Neural networks can be trained to solve partial differential equations (PDEs) by using the PDE residual as the loss function. This strategy is called "physics-informed neural networks" (PINNs), but it currently cannot produce high-accuracy…
We present a novel framework for PDE-constrained $r$-adaptivity of high-order meshes. The proposed method formulates mesh movement as an optimization problem, with an objective function defined as a convex combination of a mesh quality…
Stochastic Gradient Descent (SGD) has proven to be remarkably effective in optimizing deep neural networks that employ ever-larger numbers of parameters. Yet, improving the efficiency of large-scale optimization remains a vital and highly…
This work proposes a systematic model reduction approach based on rank adaptive tensor recovery for partial differential equation (PDE) models with high-dimensional random parameters. Since the standard outputs of interest of these models…
Imposing known physical constraints, such as conservation laws, during neural network training introduces an inductive bias that can improve accuracy, reliability, convergence, and data efficiency for modeling physical dynamics. While such…
Partial differential equation parameter estimation is a mathematical and computational process used to estimate the unknown parameters in a partial differential equation model from observational data. This paper employs a greedy sampling…
Particle filters are a frequent choice for inference tasks in nonlinear and non-Gaussian state-space models. They can either be used for state inference by approximating the filtering distribution or for parameter inference by approximating…
In high-dimensional statistics, variable selection recovers the latent sparse patterns from all possible covariate combinations. This paper proposes a novel optimization method to solve the exact L0-regularized regression problem, which is…
Recently, neural networks have been widely applied for solving partial differential equations (PDEs). Although such methods have been proven remarkably successful on practical engineering problems, they have not been shown, theoretically or…
This paper presents a novel partial differential equation (PDE)-based framework for controlling an ensemble of robots, which have limited sensing and actuation capabilities and exhibit stochastic behaviors, to perform mapping and coverage…
Design and optimal control problems are among the fundamental, ubiquitous tasks we face in science and engineering. In both cases, we aim to represent and optimize an unknown (black-box) function that associates a performance/outcome to a…
This work addresses the accurate and efficient simulation of physical phenomena governed by parametric Partial Differential Equations (PDEs) characterized by varying boundary conditions, where parametric instances modify not only the…
We consider constrained bilinear optimal control of second-order linear evolution partial differential equations (PDEs) with a reaction term on the half line, where control arises as a time-dependent reaction coefficient and constraints are…