GF-Net learns Green's functions for linear reaction-diffusion equations.
problem Learning Green's functions for linear reaction-diffusion equations on arbitrary domains.
method GF-Net, a neural network, learns Green's functions in an unsupervised manner using physics-informed approach and symmetry.
result GF-Net efficiently solves linear reaction-diffusion equations under various boundary conditions and sources.
Researchers prove inequalities for reaction-diffusion systems using a new curvature-dimension condition.
problem Proving Li-Yau and Harnack inequalities for systems of linear reaction-diffusion equations.
method Introducing a hybrid curvature-dimension condition and proving a differential Harnack estimate.
result A Harnack inequality holds under the hybrid curvature-dimension condition CDhyb(0,d) with d<∞. 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…
Study shows neural operators can efficiently solve complex reaction-diffusion systems.
problem Efficiently solving nonlinear reaction-diffusion systems using neural operators.
method Laplacian-based neural operators applied to a generalized Gierer-Meinhardt system.
result Explicit approximation error bounds established for neural operators in terms of network parameters.
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…
This study explains how adversarial interaction creates non-homogeneous patterns using a pseudo-Reaction-Diffusion model.
problem Understanding how adversarial interaction leads to non-homogeneous patterns in systems.
method Developed a pseudo-Reaction-Diffusion model to explain the mechanism.
result Turing instability is involved in creating non-homogeneous patterns.
Global solutions and smoothing effects for reaction-diffusion equations on manifolds.
problem Global existence and smoothing effects for reaction-diffusion equations on Riemannian manifolds.
method Functional analytic methods, Sobolev and Poincaré inequalities.
result Existence of global solutions under certain conditions on the manifold.
We study the phase field method for the volume preserving mean curvature flow. Given an initial C1 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.
problem Understanding reaction-diffusion equations on various manifolds.
method Analyzing the bottom of the L2 spectrum of −Δ and using time-independent nonlinearities. result Global existence of solutions for certain power nonlinearities on specific manifolds.
Global existence and smoothing effects for reaction-diffusion equations with blowup in infinite time.
problem Analyzing reaction-diffusion equations with power-type nonlinearity and slow diffusion.
method Functional analytic methods based on Sobolev and Poincaré inequalities.
result Solutions corresponding to large initial data blow up everywhere in infinite time on Cartan-Hadamard manifolds.
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.
problem Scalar curvature in generalized Ricci flow.
method Derivation of weighted scalar curvature monotonicity formulas and Perelman-type energy/entropy formulas.
result New convex Nash entropies and pseudolocality principles.
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
problem Correlation emergence and Epps effect in coupled limit order books
method Discrete random-walk description of order flow with creation, cancellation, and diffusion, coupled reaction-diffusion equations with moving reaction boundary
result Realized correlations as a function of aggregation time
Solves probabilistic Lambert problem connecting astrodynamics with optimal mass transport.
problem Determining spacecraft velocity for given positions with probabilistic constraints.
method Generalized optimal mass transport (OMT) and Schrödinger bridge (SBP) connections.
result Existence and uniqueness of solution for probabilistic Lambert problem.
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…
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…
Schrödinger bridge solved with Weyl calculus for quadratic state cost.
problem Optimal control policy to steer joint state statistics.
method Weyl calculus in quantum mechanics for reaction-diffusion PDEs.
result Explicit Markov kernel for quadratic state cost found.
New PINN architectures learn high-frequency features using Fourier features.
problem PINNs struggle with high-frequency or multi-scale features.
method Employ spatio-temporal and multi-scale random Fourier features.
result Effective PINN models for multi-scale PDEs.
Method improves simulation accuracy by mitigating distribution shift in hybrid systems.
problem Mitigating distribution shift in machine-learning augmented hybrid simulation.
method Tangent-space regularized estimator to control distribution shift.
result Marked improvements in simulation accuracy, especially for systems with high distribution shift.
New method reduces PDE model parameters by 30% with sparsity.
problem Redundant parameters in neural network projections.
method Bregman iterations for sparsity, POD compression, bias propagation.
result 30% fewer parameters with similar accuracy.
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.
problem Predicting multistable system states from limited data.
method Semi-supervised classification with SPML optimization.
result 95% accuracy in predicting reaction-diffusion equation states.
Framework uses optimal transport for neural architecture search.
problem Optimizing neural architectures in deep learning.
method Semi-discrete optimization using optimal transport.
result Gradient flow and minimizing movement scheme converge to reaction-diffusion equations.
Warped product metrics are a class of Riemannian metrics on cross products B×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.
problem Optimizing diffusion processes between given distributions.
method Regularized Schrödinger bridge with a quadratic state cost.
result Closed-form solution for the Markov kernel of the regularized Schrödinger bridge.
In this paper, we consider the heat flow for Yang-Mills connections on R5×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 …
RAD estimates gradients with less memory, faster than small batch sizes.
problem Training deep models with stochastic gradient descent requires exact gradients, but they are not needed.
method Developed a framework for randomized automatic differentiation (RAD) to compute unbiased gradient estimates with reduced memory.
result RAD converges in fewer iterations than using a small batch size for feedforward networks and similar number for recurrent networks.
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…
Revisiting Trade-sign Long-memory and Square-root Law price impact
problem Revisiting the Lillo-Mike-Farmer (LMF) theory and the square-root law (SQRL) of meta-order impact
method Using a coupled discrete reaction-diffusion formulation
result Long-memory of trade signs and square-root law of meta-order impact
PASTIS selects minimal models from stochastic dynamics data.
problem Overfitting in model selection for stochastic dynamics.
method Combining likelihood-estimation statistics with extreme value theory.
result PASTIS reliably identifies minimal models, even with low sampling rates or error.
The paper solves the Cauchy problem for Friedrichs systems on specific spacetime manifolds.
problem Investigating the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundaries.
method Admissible boundary conditions are imposed to show the existence and uniqueness of strong solutions. For hyperbolic systems, the Cauchy problem is also well-posed in the Hadamard sense.
result Existence and uniqueness of strong solutions for the Cauchy problem are proven under admissible boundary conditions.
Adaptive operator learning reduces costs in Bayesian inverse problems.
problem Reducing computational costs in Bayesian inverse problems governed by PDEs.
method Adaptive operator learning framework that gradually reduces modeling error.
result The approach significantly reduces computational costs while maintaining inversion accuracy.
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.
problem Challenging to construct ROMs from noisy high-dimensional data.
method Recurrent stochastic variational deep kernel learning (SVDKL).
result Framework accurately predicts system evolution in low-dimensional latent spaces.
Machine learning models accurately predict the state and dynamics of reactive mixing.
problem Accurate prediction of reactive mixing for Earth and environmental science applications.
method Built a high-fidelity numerical model to simulate reactive mixing scenarios. Used 20 different machine learning emulators to classify mixing state and predict three QoIs.
result Ensemble methods and MLP models accurately predict the state of reactive mixing and QoIs, significantly faster than high-fidelity simulations.
G-FuNK learns solutions for nonlinear PDEs on multiple domains and parameters.
problem Predicting time-dependent dynamics of complex systems governed by nonlinear PDEs with varying parameters and domains.
method Graph Fourier Neural Kernels combining domain-adapted and transferable components for non-diffusive and diffusive terms.
result G-FuNK achieves low relative errors on unseen domains and fiber fields, significantly accelerating predictions.
Framework augments physical models with deep learning for complex dynamics forecasting.
problem Forecasting complex dynamical phenomena with partial knowledge.
method APHYNITY framework: decomposes dynamics into physical and data-driven components.
result Framework accurately forecasts system evolution and identifies relevant parameters.
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.
problem Estimating parameters in PDEs with uncertainty quantification.
method Physics-informed neural networks (PINNs) integrated with Deep Operator Networks (DeepONets) for Bayesian inference.
result Robust and accurate solutions with comprehensive uncertainty quantification.
Develops a method for identifying structured dynamical systems from data.
problem Identifying structured dynamical systems from undersampled and noisy data.
method Sparse least-squares fitting via ℓ1−ℓ2 optimization with the alternating direction method of multipliers. result The method is stable and successful under certain conditions, as shown by theoretical guarantees and computational results.
GD-VAEs learn dynamics from observations using geometric and topological information.
problem Learning parsimonious representations of nonlinear dynamics from observations.
method Develops data-driven methods incorporating geometric and topological information using Variational Autoencoders (VAEs).
result GD-VAEs provide methods for learning reduced dimensional representations of nonlinear dynamics.
One of the open problems in scientific computing is the long-time integration of nonlinear stochastic partial differential equations (SPDEs). We address this problem by taking advantage of recent advances in scientific machine learning and the dynamically orthogonal (DO) and bi-orthogonal (BO) methods for representing …
New method solves high-dimensional Bayesian inverse problems efficiently.
problem Efficiently solving high-dimensional Bayesian inverse problems with limited data.
method Physics-informed Neural Operators with RealNVP architecture for invertibility and differentiability.
result Accurate approximations of the full posterior without additional forward solves or sampling.
Neural operators correct PDE residuals to improve BIP solutions.
problem Reducing error in infinite-dimensional Bayesian inverse problems with neural operators.
method Error correction using PDE residuals to improve neural operator approximation.
result Trained neural operators with error correction achieve a quadratic reduction in approximation error.