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Physics-Informed Neural Networks (PINNs) seek to solve partial differential equations (PDEs) with deep learning. Mainstream approaches that deploy fully-connected multi-layer deep learning architectures require prolonged training to achieve…
Species transport models typically combine partial differential equations (PDEs) with relations from hindered transport theory to quantify electromigrative, convective, and diffusive transport through complex nanoporous systems; however,…
The least squares method with deep neural networks as function parametrization has been applied to solve certain high-dimensional partial differential equations (PDEs) successfully; however, its convergence is slow and might not be…
Physics-informed neural networks (PINNs) have recently become a popular method for solving forward and inverse problems governed by partial differential equations (PDEs). By incorporating the residual of the PDE into the loss function of a…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
We present a data-driven control framework for partial differential equations (PDEs). Our approach integrates Time-Integrated Deep Operator Networks (TI-DeepONets) as differentiable PDE surrogate models within the Differentiable Predictive…
In many science and engineering settings, system dynamics are characterized by governing PDEs, and a major challenge is to solve inverse problems (IPs) where unknown PDE parameters are inferred based on observational data gathered under…
The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…
Partial Differential Equations (PDEs) are central to modeling complex systems across physical, biological, and engineering domains, yet traditional numerical methods often struggle with high-dimensional or complex problems. Physics-Informed…
An innovative physics-guided learning algorithm for predicting the mechanical response of materials and structures is proposed in this paper. The key concept of the proposed study is based on the fact that physics models are governed by…
We propose a trainable-by-parts surrogate model for solving forward and inverse parameterized nonlinear partial differential equations. Like several other surrogate and operator learning models, the proposed approach employs an encoder to…
Deep Operator Networks (DeepONets) have recently emerged as powerful data-driven frameworks for learning nonlinear operators, particularly suited for approximating solutions to partial differential equations. Despite their promising…
Partial differential equations (PDEs) play a central role in describing many physical phenomena. Various scientific and engineering applications demand a versatile and differentiable PDE solver that can quickly generate solutions with…
Solving inverse problems governed by partial differential equations (PDEs) is central to science and engineering, yet remains challenging when measurements are sparse, noisy, or when the underlying coefficients are high-dimensional or…
I will demonstrate the effectiveness of Physics-Informed Neural Networks (PINNs) in solving partial differential equations (PDEs) when training data are scarce or noisy. The training data can be located either at the boundaries or within…
Identifying parameters in partial differential equations (PDEs) represents a very broad class of applied inverse problems. In recent years, several unsupervised learning approaches using (deep) neural networks have been developed to solve…
Mixed-dimensional partial differential equations (PDEs) are characterized by coupled operators defined on domains of varying dimensions and pose significant computational challenges due to their inherent ill-conditioning. Moreover, the…
We present a new data-driven reduced-order modeling approach to efficiently solve parametrized partial differential equations (PDEs) for many-query problems. This work is inspired by the concept of implicit neural representation (INR),…
Systems modeled by partial differential equations (PDEs) are at least as ubiquitous as systems that are by nature finite-dimensional and modeled by ordinary differential equations (ODEs). And yet, systematic and readily usable…
Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately…