Related papers: Closure Discovery for Coarse-Grained Partial Diffe…
Deep learning has been highly successful in some applications. Nevertheless, its use for solving partial differential equations (PDEs) has only been of recent interest with current state-of-the-art machine learning libraries, e.g.,…
Foundation models for partial differential equations (PDEs) have emerged as powerful surrogates pre-trained on diverse physical systems, but adapting them to new downstream tasks remains challenging due to limited task-specific data and…
Discovering governing Partial Differential Equations (PDEs) from sparse and noisy data is a challenging issue in data-driven scientific computing. Conventional sparse regression methods often suffer from two major limitations: (i) the…
The combination of high-dimensionality and disparity of time scales encountered in many problems in computational physics has motivated the development of coarse-grained (CG) models. In this paper, we advocate the paradigm of data-driven…
We present a novel model Graph Neural Stochastic Differential Equations (Graph Neural SDEs). This technique enhances the Graph Neural Ordinary Differential Equations (Graph Neural ODEs) by embedding randomness into data representation using…
Systems involving Partial Differential Equations (PDEs) have recently become more popular among the machine learning community. However prior methods usually treat infinite dimensional problems in finite dimensions with Reduced Order…
Many scientific and industrial applications require solving Partial Differential Equations (PDEs) to describe the physical phenomena of interest. Some examples can be found in the fields of aerodynamics, astrodynamics, combustion and many…
We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the…
Physics-informed neural network (PINN) is a powerful emerging method for studying forward-inverse problems of partial differential equations (PDEs), even from limited sample data. Variable coefficient PDEs, which model real-world phenomena,…
Although deep-learning has been successfully applied in a variety of science and engineering problems owing to its strong high-dimensional nonlinear mapping capability, it is of limited use in scientific knowledge discovery. In this work,…
We combine concepts from multilevel solvers for partial differential equations (PDEs) with neural network based deep learning and propose a new methodology for the efficient numerical solution of high-dimensional parametric PDEs. An…
In many areas, such as the physical sciences, life sciences, and finance, control approaches are used to achieve a desired goal in complex dynamical systems governed by differential equations. In this work we formulate the problem of…
Discovering hidden partial differential equations (PDEs) and operators from data is an important topic at the frontier between machine learning and numerical analysis. This doctoral thesis introduces theoretical results and deep learning…
We consider solving complex spatiotemporal dynamical systems governed by partial differential equations (PDEs) using frequency domain-based discrete learning approaches, such as Fourier neural operators. Despite their widespread use for…
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…
Recent research works for solving partial differential equations (PDEs) with deep neural networks (DNNs) have demonstrated that spatiotemporal function approximators defined by auto-differentiation are effective for approximating nonlinear…
A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization…
Climate models are essential to understand and project climate change, yet long-standing biases and uncertainties in their projections remain. This is largely associated with the representation of subgrid-scale processes, particularly…
Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, with solutions over the…
Physics-informed neural networks (PINNs) have lately received significant attention as a representative deep learning-based technique for solving partial differential equations (PDEs). Most fully connected network-based PINNs use automatic…