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Physics-informed neural networks (PINNs) have recently emerged as an alternative way of solving partial differential equations (PDEs) without the need of building elaborate grids, instead, using a straightforward implementation. In…
Stochastic partial differential equations (SPDEs) are ubiquitous in engineering and computational sciences. The stochasticity arises as a consequence of uncertainty in input parameters, constitutive relations, initial/boundary conditions,…
Learning operators from data is central to scientific machine learning. While DeepONets are widely used for their ability to handle complex domains, they require fixed sensor numbers and locations, lack mechanisms for uncertainty…
We develop a probabilistic machine learning method, which formulates a class of stochastic neural networks by a stochastic optimal control problem. An efficient stochastic gradient descent algorithm is introduced under the stochastic…
Stochastic partial differential equations (SPDEs) represent a very active research field with numerous recent developments and breakthrough results. There are several well-established approaches and methods used to construct solutions for…
Operator learning has emerged as a powerful tool in scientific computing for approximating mappings between infinite-dimensional function spaces. A primary application of operator learning is the development of surrogate models for the…
Neural operators aim to approximate the solution operator of a system of differential equations purely from data. They have shown immense success in modeling complex dynamical systems across various domains. However, the occurrence of…
Uncertainty quantification appears today as a crucial point in numerous branches of science and engineering. In the past two decades, a growing interest has been devoted to stochastic finite element method (SFEM) for the propagation of…
We propose a novel \textit{capsule} based deep encoder-decoder model for surrogate modeling and uncertainty quantification of systems in mechanics from sparse data. The proposed framework is developed by adapting Capsule Network (CapsNet)…
We propose a novel framework for adaptively learning the time-evolving solutions of stochastic partial differential equations (SPDEs) using score-based diffusion models within a recursive Bayesian inference setting. SPDEs play a central…
In recent years, deep learning for modeling physical phenomena which can be described by partial differential equations (PDEs) have received significant attention. For example, for learning Hamiltonian mechanics, methods based on deep…
Learning solution operators for differential equations with neural networks has shown great potential in scientific computing, but ensuring their stability under input perturbations remains a critical challenge. This paper presents a robust…
Differential equations are widely used to describe complex dynamical systems with evolving parameters in nature and engineering. Effectively learning a family of maps from the parameter function to the system dynamics is of great…
I propose a novel framework that integrates stochastic differential equations (SDEs) with deep generative models to improve uncertainty quantification in machine learning applications involving structured and temporal data. This approach,…
Stochastic differential equations (SDEs) are a ubiquitous modeling framework that finds applications in physics, biology, engineering, social science, and finance. Due to the availability of large-scale data sets, there is growing interest…
Solving Singularly Perturbed Differential Equations (SPDEs) poses computational challenges arising from the rapid transitions in their solutions within thin regions. The effectiveness of deep learning in addressing differential equations…
Neural operators (NOs) provide a new paradigm for efficiently solving partial differential equations (PDEs), but their training depends on costly high-fidelity data from numerical solvers, limiting applications in complex systems. We…
Recent advances in the theory of Neural Operators (NOs) have enabled fast and accurate computation of the solutions to complex systems described by partial differential equations (PDEs). Despite their great success, current NO-based…
We present a new scientific machine learning method that learns from data a computationally inexpensive surrogate model for predicting the evolution of a system governed by a time-dependent nonlinear partial differential equation (PDE), an…
A new data-driven method for operator learning of stochastic differential equations(SDE) is proposed in this paper. The central goal is to solve forward and inverse stochastic problems more effectively using limited data. Deep operator…