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This paper proposes a methodology to estimate stress in the subsurface by a hybrid method combining finite element modeling and neural networks. This methodology exploits the idea of obtaining a multi-frequency solution in the numerical…
We combine conditional state density construction with an extension of the Scenario Approach for stochastic Model Predictive Control to nonlinear systems to yield a novel particle-based formulation of stochastic nonlinear output-feedback…
In this paper, we introduce and study one-dimensional models for the behavior of pedestrians in a narrow street or corridor. We begin at the microscopic level by formulating a stochastic cellular automata model with explicit rules for…
The brain-body-environment framework studies adaptive behavior through embodied and situated agents, emphasizing interactions between brains, biomechanics, and environmental dynamics. However, many models often treat the brain as a network…
Reaction-diffusion processes are the foundational model for a diverse range of complex systems, ranging from biochemical reactions to social agent-based phenomena. The underlying dynamics of these systems occur at the individual…
The numerical solution of high dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid--based methods impractical beyond a few dimensions. In recent years,…
In this work, we derive particle schemes, based on micro-macro decomposition, for linear kinetic equations in the diffusion limit. Due to the particle approximation of the micro part, a splitting between the transport and the collision part…
The modelling and analysis of biological systems has deep roots in Mathematics, specifically in the field of ordinary differential equations (ODEs). Alternative approaches based on formal calculi, often derived from process algebras or term…
This paper presents a two-phase method for learning interaction kernels of stochastic many-particle systems. After transforming stochastic trajectories of every particle into the particle density function by the kernel density estimation…
This paper explores the efficacy of diffusion-based generative models as neural operators for partial differential equations (PDEs). Neural operators are neural networks that learn a mapping from the parameter space to the solution space of…
Analysis of large continuous-time stochastic systems is a computationally intensive task. In this work we focus on population models arising from chemical reaction networks (CRNs), which play a fundamental role in analysis and design of…
Rendering highly scattering participating media using brute force path tracing is a challenge. The diffusion approximation reduces the problem to solving a simple linear partial differential equation. Flux-limited diffusion introduces…
[Context] The stochasticity of grain chemistry requires special care in modeling. Previously methods based on the modified rate equation, the master equation, the moment equation, and Monte Carlo simulations have been used. [Aims] We…
Numerical resolution of high-dimensional nonlinear PDEs remains a huge challenge due to the curse of dimensionality. Starting from the weak formulation of the Lawson-Euler scheme, this paper proposes a stochastic particle method (SPM) by…
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…
Nowadays, neural networks are widely used in many applications as artificial intelligence models for learning tasks. Since typically neural networks process a very large amount of data, it is convenient to formulate them within the…
It has recently been shown that structural conditions on the reaction network, rather than a 'fine-tuning' of system parameters, often suffice to impart 'absolute concentration robustness' on a wide class of biologically relevant,…
Modeling the dynamics of deformable objects is challenging due to their diverse physical properties and the difficulty of estimating states from limited visual information. We address these challenges with a neural dynamics framework that…
In this paper, we study multi-species stochastic interacting particle systems and their mean-field McKean-Vlasov partial differential equations (PDEs) in non-convex landscapes. We discuss the well-posedness of the multi-species SDE system,…
In this paper we introduce a class of stochastic population models based on "patch dynamics". The size of the patch may be varied, and this allows one to quantify the departures of these stochastic models from various mean field theories,…