Related papers: Bayesian neural networks for weak solution of PDEs…
Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for…
Neural networks (NNs) are primarily developed within the frequentist statistical framework. Nevertheless, frequentist NNs lack the capability to provide uncertainties in the predictions, and hence their robustness can not be adequately…
Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…
Physics-informed neural networks (PINNs) have emerged as a promising approach to solving partial differential equations (PDEs) using neural networks, particularly in data-scarce scenarios, due to their unsupervised training capability.…
Conventional Bayesian Neural Networks (BNNs) are unable to leverage unlabelled data to improve their predictions. To overcome this limitation, we introduce Self-Supervised Bayesian Neural Networks, which use unlabelled data to learn models…
Multiscale and multiphysics problems need novel numerical methods in order for them to be solved correctly and predictively. To that end, we develop a wavelet based technique to solve a coupled system of nonlinear partial differential…
The discovery of Partial Differential Equations (PDEs) is an essential task for applied science and engineering. However, data-driven discovery of PDEs is generally challenging, primarily stemming from the sensitivity of the discovered…
We present a methodology combining neural networks with physical principle constraints in the form of partial differential equations (PDEs). The approach allows to train neural networks while respecting the PDEs as a strong constraint in…
In recent years, there has been a growing interest in leveraging deep learning and neural networks to address scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods…
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 introduce a new class of spatially stochastic physics and data informed deep latent models for parametric partial differential equations (PDEs) which operate through scalable variational neural processes. We achieve this by assigning…
In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasilinear parabolic partial differential equations (PDEs),…
Compared to point estimates calculated by standard neural networks, Bayesian neural networks (BNN) provide probability distributions over the output predictions and model parameters, i.e., the weights. Training the weight distribution of a…
Neural networks are universal approximators and are studied for their use in solving differential equations. However, a major criticism is the lack of error bounds for obtained solutions. This paper proposes a technique to rigorously…
In recent years, deep learning has proven to be a viable methodology for surrogate modeling and uncertainty quantification for a vast number of physical systems. However, in their traditional form, such models can require a large amount of…
Constrained optimization problems arise in various engineering systems such as inventory management and power grids. Standard deep neural network (DNN) based machine learning proxies are ineffective in practical settings where labeled data…
In the realm of computational fluid dynamics, traditional numerical methods, which heavily rely on discretization, typically necessitate the formulation of partial differential equations (PDEs) in conservative form to accurately capture…
This work proposes a solution for the problem of training physics-informed networks under partial integro-differential equations. These equations require an infinite or a large number of neural evaluations to construct a single residual for…
This paper proposes a sparse Bayesian treatment of deep neural networks (DNNs) for system identification. Although DNNs show impressive approximation ability in various fields, several challenges still exist for system identification…
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…