arXiv research

Study shows neural operators can efficiently solve complex reaction-diffusion systems.

Researchers prove inequalities for reaction-diffusion systems using a new curvature-dimension condition.

Complex behaviour in many systems arises from the stochastic interactions of spatially distributed particles or agents. Stochastic reaction-diffusion processes are widely used to model such behaviour in disciplines ranging from biology to the social sciences, yet they are notoriously difficult to simulate and calibrate…

We introduce and illustrate a new approach to the unknotting problem via the dynamics of vortex strings in a nonlinear partial differential equation of reaction-diffusion type. To untangle a given knot, a Biot-Savart construction is used to initialize the knot as a vortex string in the FitzHugh-Nagumo equation. Remarka…

GF-Net learns Green's functions for linear reaction-diffusion equations.

This study explains how adversarial interaction creates non-homogeneous patterns using a pseudo-Reaction-Diffusion model.

Global solutions and smoothing effects for reaction-diffusion equations on manifolds.

We study the phase field method for the volume preserving mean curvature flow. Given an initial $C^1$ hypersurface we proved the existence of the weak solution for the volume preserving mean curvature flow via the reaction diffusion equation with a nonlocal term. We also show the monotonicity formula and the density up…

Global solutions found for certain reaction-diffusion equations on specific manifolds.

We generalize the reaction-diffusion model A + B -> 0 in order to study the impact of an excess of A (or B) at the reaction front. We provide an exact solution of the model, which shows that linear response breaks down: the average displacement of the reaction front grows as the square-root of the imbalance. We argue t…

Global existence and smoothing effects for reaction-diffusion equations with blowup in infinite time.

In this paper we construct a parametrization-free embedding technique for numerically evolving reaction-diffusion PDEs defined on algebraic curves that possess an isolated singularity. In our approach, we first desingularize the curve by appealing to techniques from algebraic geometry. We create a family of smooth curv…

Several models of stock trading [P. Bak et al, Physica A {\bf 246}, 430 (1997)] are analyzed in analogy with one-dimensional, two-species reaction-diffusion-branching processes. Using heuristic and scaling arguments, we show that the short-time market price variation is subdiffusive with a Hurst exponent $H=1/4$. Biase…

New formulas derived for scalar curvature in generalized Ricci flow.

In this note, we propose Bernstein's problem and De Giorgi's conjecture for spatially inhomogeneous equations, as well as De Giorgi's conjecture for system of reaction-diffusion equations.

Unified analytic account of correlation emergence and Epps effect in coupled limit order books

Solves probabilistic Lambert problem connecting astrodynamics with optimal mass transport.

We suggest that the broad distribution of time scales in financial markets could be a crucial ingredient to reproduce realistic price dynamics in stylised Agent-Based Models. We propose a fractional reaction-diffusion model for the dynamics of latent liquidity in financial markets, where agents are very heterogeneous i…

We use probabilistic methods to study classical solutions for systems of interacting semilinear parabolic partial differential equations. In a modeling framework for a financial market with interacting Ito and point processes, such PDEs are shown to provide a natural description for the solution of hedging and valuatio…

Schrödinger bridge solved with Weyl calculus for quadratic state cost.

New PINN architectures learn high-frequency features using Fourier features.

We extend the viscosity solution characterization proved in [5] for call/put American option prices to the case of a general payoff function in a multi-dimensional setting: the price satisfies a semilinear re-action/diffusion type equation. Based on this, we propose two new numerical schemes inspired by the branching p…

Method predicts multistable system states from sparse measurements.

Framework uses optimal transport for neural architecture search.

Warped product metrics are a class of Riemannian metrics on cross products $B \times F$ which have been well studied and provide a rich set of examples. In this paper we consider shrinking gradient Ricci solitons which are warped product metrics. We prove that if the curvature of the metric is bounded and the base $B$ …

We solve a Schrödinger bridge with a quadratic state cost, finding a closed-form solution.

In this paper, we consider the heat flow for Yang-Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)-$equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove \cite{Wei04}. We prove …

We study a class of nonlocal, energy-driven dynamical models that govern the motion of closed, embedded curves from both an energetic and dynamical perspective. Our energetic results provide a variety of ways to understand physically motivated energetic models in terms of more classical, combinatorial measures of compl…

PASTIS selects minimal models from stochastic dynamics data.

The paper solves the Cauchy problem for Friedrichs systems on specific spacetime manifolds.

A microscopic approach to macroeconomic features is intended. A model for macroeconomic behavior under heterogeneous spatial economic conditions is reviewed. A birth-death lattice gas model taking into account the influence of an economic environment on the fitness and concentration evolution of economic entities is nu…

Method learns latent dynamics of complex systems from noisy data.

We present an extended version of the recently proposed "LLOB" model for the dynamics of latent liquidity in financial markets. By allowing for finite cancellation and deposition rates within a continuous reaction-diffusion setup, we account for finite memory effects on the dynamics of the latent order book. We compute…

It has been observed that residual networks can be viewed as the explicit Euler discretization of an Ordinary Differential Equation (ODE). This observation motivated the introduction of so-called Neural ODEs, which allow more general discretization schemes with adaptive time stepping. Here, we propose ANODEV2, which is…

DeepONets combine neural networks with physics constraints for PDEs and parameter estimation.

Develops a method for identifying structured dynamical systems from data.

GD-VAEs learn dynamics from observations using geometric and topological information.

New method solves high-dimensional Bayesian inverse problems efficiently.

Form a pure mathematical point of view, common functional forms representing different physical phenomena can be defined. For example, rates of chemical reactions, diffusion and heat transfer are all governed by exponential-type expressions. If machine learning is used for physical problems, inferred from domain knowle…

New method extracts hidden phases in binary mixtures using tubular tilings.

Method improves simulation accuracy by mitigating distribution shift in hybrid systems.

INP accelerates stochastic simulations using deep Bayesian active learning.

CViT learns complex physical systems using vision transformer techniques.

The regulatory process of Drosophila is thoroughly studied for understanding a great variety of biological principles. While pattern-forming gene networks are analysed in the transcription step, post-transcriptional events (e.g. translation, protein processing) play an important role in establishing protein expression …

Compositional diffusion models simulate coupled PDEs efficiently.

This paper presents an efficient algorithm for evolving point cloud data on smooth manifolds using B-Splines.

New method reduces PDE model parameters by 30% with sparsity.

The \textit{parabolic scalar curvature equation} is a reaction-diffusion type equation on an $(n-1)$-manifold $Σ$, the time variable of which shall be denoted by $r$. Given a function $R$ on $[r_0,r_1)\timesΣ$ and a family of metrics $γ(r)$ on $Σ$, when the coefficients of this equation are appropriately defined in ter…