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We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…
Physics-informed machine learning (PIML) has emerged as a promising alternative to conventional numerical methods for solving partial differential equations (PDEs). PIML models are increasingly built via deep neural networks (NNs) whose…
The first order by time partial differential equations are used as models in applications such as fluid flow, heat transfer, solid deformation, electromagnetic waves, and others. In this paper we propose the new numerical method to solve a…
This paper proposes a hierarchical, multi-resolution framework for the identification of model parameters and their spatially variability from noisy measurements of the response or output. Such parameters are frequently encountered in…
Elliptic Partial Differential Equations (PDEs) play a central role in computing the equilibrium conditions of physical problems (heat, gravitation, electrostatics, etc.). Efficient solutions to elliptic PDEs are also relevant to computer…
We introduce variational spectral learning (VSL), a machine learning framework for solving partial differential equations (PDEs) that operates directly in the coefficient space of spectral expansions. VSL offers a principled bridge between…
We present a discretization-free scalable framework for solving a large class of mass-conserving partial differential equations (PDEs), including the time-dependent Fokker-Planck equation and the Wasserstein gradient flow. The main…
We introduce a generative learning framework to model high-dimensional parametric systems using gradient guidance and virtual observations. We consider systems described by Partial Differential Equations (PDEs) discretized with structured…
Partial differential equations (PDEs) govern nearly every physical process in science and engineering, yet solving them at scale remains prohibitively expensive. Generative AI has transformed language, vision, and protein science, but…
Partial differential equation (PDE) models are widely used in engineering and natural sciences to describe spatio-temporal processes. The parameters of the considered processes are often unknown and have to be estimated from experimental…
Modeling complex spatiotemporal dynamical systems, such as the reaction-diffusion processes, have largely relied on partial differential equations (PDEs). However, due to insufficient prior knowledge on some under-explored dynamical…
We study a quantum-algorithmic framework for parameterizing partial differential equations (PDEs). For a broad class of problems in which the discretized parameter field admits a diagonal representation, block-encodings of diagonal…
We are interested in the approximation of partial differential equations with a data-driven approach based on the reduced basis method and machine learning. We suppose that the phenomenon of interest can be modeled by a parametrized partial…
Partial differential equations (PDEs) are ubiquitous in the world around us, modelling phenomena from heat and sound to quantum systems. Recent advances in deep learning have resulted in the development of powerful neural solvers; however,…
Starting with sets of disorganized observations of spatially varying and temporally evolving systems, obtained at different (also disorganized) sets of parameters, we demonstrate the data-driven derivation of parameter dependent,…
Many systems in physics, engineering, and biology exhibit multiscale stochastic dynamics, where low-dimensional slow variables evolve under the influence of high-dimensional fast processes. In practice, observations are often limited to a…
This paper is interested in developing reduced order models (ROMs) for repeated simulation of fractional elliptic partial differential equations (PDEs) for multiple values of the parameters (e.g., diffusion coefficients or fractional…
Methods for solving PDEs using neural networks have recently become a very important topic. We provide an a priori error analysis for such methods which is based on the $\mathcal{K}_1(\mathbb{D})$-norm of the solution. We show that the…
We present a framework for recovering/approximating unknown time-dependent partial differential equation (PDE) using its solution data. Instead of identifying the terms in the underlying PDE, we seek to approximate the evolution operator of…
The modern machine learning methods allow one to obtain the data-driven models in various ways. However, the more complex the model is, the harder it is to interpret. In the paper, we describe the algorithm for the mathematical equations…