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Learning dynamical systems from incomplete or noisy data is inherently ill-posed, as a single observation may correspond to multiple plausible futures. While physics-based ensemble forecasting relies on perturbing initial states to capture…
We introduce and analyze a method of learning-informed parameter identification for partial differential equations (PDEs) in an all-at-once framework. The underlying PDE model is formulated in a rather general setting with three unknowns:…
In this article, we introduce a system of stochastic differential equations (SDEs) consisting of time-dependent covariates and consider both fixed and random effects set-ups. We also allow the functional part associated with the drift…
Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often…
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…
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…
Given multiple images that describe chaotic reaction-diffusion dynamics, parameters of a PDE model are estimated using autosynchronization, where parameters are controlled by synchronization of the model to the observed data. A…
We consider the problem of solving partial differential equations (PDEs) in domains with complex microparticle geometry that is impractical, or intractable, to model explicitly. Drawing inspiration from volume rendering, we propose tackling…
This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe's method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an…
Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural…
The advection-diffusion and wave equations are the fundamental equations governing any physical law and therefore arise in many areas of physics and astrophysics. For complex problems and geometries, only numerical simulations can give…
Time-varying stochastic optimization problems frequently arise in machine learning practice (e.g. gradual domain shift, object tracking, strategic classification). Although most problems are solved in discrete time, the underlying process…
We propose an alternative method for one-dimensional continuum diffusion models with spatially variable (heterogeneous) diffusivity. Our method, which extends recent work on stochastic diffusion, assumes the constant-coefficient homogenized…
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…
Recent advances in stochastic differential equations (SDEs) have enabled robust modeling of real-world dynamical processes across diverse domains, such as finance, health, and systems biology. However, parameter estimation for SDEs…
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,…
Presence of a high-dimensional stochastic parameter space with discontinuities poses major computational challenges in analyzing and quantifying the effects of the uncertainties in a physical system. In this paper, we propose a stochastic…
We establish stable finite element (FE) approximations of convection-diffusion initial boundary value problems using the automatic variationally stable finite element (AVS-FE) method. The transient convection-diffusion problem leads to…
We propose DiffusionRollout, a novel selective rollout planning strategy for autoregressive diffusion models, aimed at mitigating error accumulation in long-horizon predictions of physical systems governed by partial differential equations…
Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…