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We introduce a general framework for solving partial differential equations (PDEs) using generative diffusion models. In particular, we focus on the scenarios where we do not have the full knowledge of the scene necessary to apply classical…
Solving partial differential equations (PDEs) on fine spatio-temporal scales for high-fidelity solutions is critical for numerous scientific breakthroughs. Yet, this process can be prohibitively expensive, owing to the inherent complexities…
We present Mechanistic PDE Networks -- a model for discovery of governing partial differential equations from data. Mechanistic PDE Networks represent spatiotemporal data as space-time dependent linear partial differential equations in…
Partial differential equations (PDEs) with multiple scales or those defined over sufficiently large domains arise in various areas of science and engineering and often present problems when approximating the solutions numerically. Machine…
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…
Stochastic solutions provide new rigorous results for nonlinear PDE's and, through its local non-grid nature, are a natural tool for parallel computation. There are two different approaches for the construction of stochastic solutions:…
In this paper, we develop an ensemble-based time-stepping algorithm to efficiently find numerical solutions to a group of linear, second-order parabolic partial differential equations (PDEs). Particularly, the PDE models in the group could…
Large-scale optimization problems that involve thousands of decision variables have extensively arisen from various industrial areas. As a powerful optimization tool for many real-world applications, evolutionary algorithms (EAs) fail to…
Modeling real-world problems with partial differential equations (PDEs) is a prominent topic in scientific machine learning. Classic solvers for this task continue to play a central role, e.g. to generate training data for deep learning…
We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate…
The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length and timescales. Often, it is computationally intractable to resolve the finest features…
We extend a recently developed method to solve semi-linear PDEs to the case of a degenerated diffusion. Being a pure Monte Carlo method it does not suffer from the so called curse of dimensionality and it can be used to solve problems that…
What is a systematic way to efficiently apply a wide spectrum of advanced ML programs to industrial scale problems, using Big Models (up to 100s of billions of parameters) on Big Data (up to terabytes or petabytes)? Modern parallelization…
Physical models with uncertain inputs are commonly represented as parametric partial differential equations (PDEs). That is, PDEs with inputs that are expressed as functions of parameters with an associated probability distribution.…
Many high-dimensional uncertainty quantification problems are solved by polynomial dimensional decomposition (PDD), which represents Fourier-like series expansion in terms of random orthonormal polynomials with increasing dimensions. This…
We introduce a guided stochastic sampling method that augments sampling from diffusion models with physics-based guidance derived from partial differential equation (PDE) residuals and observational constraints, ensuring generated samples…
In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating PDEs with two characteristic scales. From a continuous perspective, our formulation…
Common techniques for the spatial discretisation of PDEs on a macroscale grid include finite difference, finite elements and finite volume methods. Such methods typically impose assumed microscale structures on the subgrid fields, so…
Stochastic gradient descent (SGD) is a widely adopted iterative method for optimizing differentiable objective functions. In this paper, we propose and discuss a novel approach to scale up SGD in applications involving non-convex functions…
This paper proposes a domain decomposition subspace neural network method for efficiently solving linear and nonlinear partial differential equations. By combining the principles of domain decomposition and subspace neural networks, the…