Related papers: Questionnaires to PDEs: From Disorganized Data to …
Deriving governing equations in Electromagnetic (EM) environment based on first principles can be quite tough when there are some unknown sources of noise and other uncertainties in the system. For nonlinear multiple-physics electromagnetic…
Partial differential equations (PDEs) are a central tool for modeling the dynamics of physical, engineering, and materials systems, but high-fidelity simulations are often computationally expensive. At the same time, many scientific…
Scenario-based probabilistic forecasts have become vital for decision-makers in handling intermittent renewable energies. This paper presents a recent promising deep learning generative approach called denoising diffusion probabilistic…
Score-based diffusion models have emerged as a powerful class of generative methods, achieving state-of-the-art performance across diverse domains. Despite their empirical success, the mathematical foundations of those models remain only…
Spatial reaction-diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique most commonly adopted in the literature implements systems of partial differential equations (PDEs),…
The Transformer architecture has revolutionized artificial intelligence, yet a principled theoretical understanding of its internal mechanisms remains elusive. This paper introduces a novel analytical framework that reconceptualizes the…
Solving partial differential equations (PDEs) can be prohibitively expensive using traditional numerical methods. Deep learning-based surrogate models typically specialize in a single PDE with fixed parameters. We present a framework for…
Modeling the traffic dynamics is essential for understanding and predicting the traffic spatiotemporal evolution. However, deriving the partial differential equation (PDE) models that capture these dynamics is challenging due to their…
Despite their claimed biological plausibility, most self organizing networks have strict topological constraints and consequently they cannot take into account a wide range of external stimuli. Furthermore their evolution is conditioned by…
Projection-based reduced order models are effective at approximating parameter-dependent differential equations that are parametrically separable. When parametric separability is not satisfied, which occurs in both linear and nonlinear…
Geological parameterization entails the representation of a geomodel using a small set of latent variables and a mapping from these variables to grid-block properties such as porosity and permeability. Parameterization is useful for data…
Understanding how species persist under interacting stressors is a central challenge in ecology. We develop a spatially explicit reaction-diffusion framework to investigate competing species in landscapes shaped by climate variability,…
Usually, the systems of partial differential equations (PDEs) are discovered from observational data in the single vector equation form. However, this approach restricts the application to the real cases, where, for example, the form of the…
Diffusion models have been widely studied as effective generative tools for solving inverse problems. The main ideas focus on performing the reverse sampling process conditioned on noisy measurements, using well-established numerical…
We present a method of discovering governing differential equations from data without the need to specify a priori the terms to appear in the equation. The input to our method is a dataset (or ensemble of datasets) corresponding to a…
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…
A potent class of generative models known as Diffusion Probabilistic Models (DPMs) has become prominent. A forward diffusion process adds gradually noise to data, while a model learns to gradually denoise. Sampling from pre-trained DPMs is…
Time-dependent Partial Differential Equations with given initial conditions are considered in this paper. New differentiation techniques of the unknown solution with respect to time variable are proposed. It is shown that the proposed…
We propose a machine learning framework to accelerate numerical computations of time-dependent ODEs and PDEs. Our method is based on recasting (generalizations of) existing numerical methods as artificial neural networks, with a set of…
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…