arXiv research

PINNs struggle with data-to-PDE inconsistencies, limiting their accuracy.

This paper is concerned with the following Markovian stochastic differential equation of mean-reversion type \[ dR_t= (θ+σα(R_t, t))R_t dt +σR_t dB_t \] with an initial value $R_0=r_0\in\mathbb{R}$, where $θ\in\mathbb{R}$ and $σ>0$ are constants, and the mean correction function $α:\mathbb{R}\times[0,\infty)\to α(x,t)\…

The paper analyzes the score field of diffusion models using Burgers dynamics.

A new method predicts non-Markovian closure terms for complex systems.

In recent years, deep learning has proven to be a viable methodology for surrogate modeling and uncertainty quantification for a vast number of physical systems. However, in their traditional form, such models can require a large amount of training data. This is of particular importance for various engineering and scie…

RandNet-Parareal uses neural networks to speed up time-parallel PDE solving.

Develops a method for identifying structured dynamical systems from data.

Develops neural network approximations for infinite-dimensional input-output maps.

We present a numerical model for the dynamics of thin viscous threads based on a discrete, Lagrangian formulation of the smooth equations. The model makes use of a condensed set of coordinates, called the centerline/spin representation: the kinematical constraints linking the centerline's tangent to the orientation of …

This study evaluates the importance of design of experiments for PINN in physics-informed deep learning.

Paper introduces mcTangent for real-time dynamical systems.

On curved spaces, viscous fluids reach equilibrium quickly.

Bayesian method learns PDEs from noisy data.

The paper derives the QGS equations using stochastic central extensions.

In this paper, using Riemann-Lagrange geometrical methods, we construct a geometrical model on 1-jet spaces for the study of multi-time relativistic magnetized non-viscous plasma, characterized by a given energy-stress-momentum distinguished (d-) tensor. In that arena, we give the conservation laws and the continuity e…

Using a simple and well-motivated modification of the stress-energy tensor for a viscous fluid proposed by Lichnerowicz, we prove that Einstein's equations coupled to a relativistic version of the Navier-Stokes equations are well-posed in a suitable Gevrey class if the fluid is incompressible and irrotational. These la…

WSINDy algorithm proves robust to noise in identifying differential equations.

Model reduction methods aim to describe complex dynamic phenomena using only relevant dynamical variables, decreasing computational cost, and potentially highlighting key dynamical mechanisms. In the absence of special dynamical features such as scale separation or symmetries, the time evolution of these variables typi…

Let $f,g:{\Bbb R}\to{\Bbb R}$ be integrable functions, $f$ nowhere zero, and $φ(u)=\int du/f(u)$ be invertible. An exact solution to the generalized nonhomogeneous inviscid Burgers' equation $u_t+g(u).u_x=f(u)$ is given, by quadratures.

We establish a simple relation between curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation corresponds to a focal point of the group of volume-preserving diffeomorphisms regard…

Random feature model approximates PDE solutions efficiently.

Continuous functions on graphs in Carnot groups satisfy a Burgers' type equation.

Enhanced autoencoders improve ROMs for PDEs by capturing essential properties.

New approach connects UQ in SciML to viscous HJ PDEs for efficient uncertainty quantification.

Preliminary group classification for a class of generalized inviscid Burger's equations in the general form $u_t+g(x, u)u_x = f(x, u)$ is given and additional equivalence transformations are found. Adduced results complete and essentially generalize recent works on the subject . A number of new interesting nonlinear in…

In this paper we consider the log-aesthetic curves and their generalization which are used in CAGD. We consider those curves under similarity geometry and characterize them as stationary integrable flow on plane curves which is governed by the Burgers equation. We propose a variational formulation of those curves whose…

New method solves PDEs for any initial condition without retraining.

The paper proves that any asymptotically shearfree congruence at the conformal infinity (scri) in a (2,2)-signature spacetime is determined locally by a solution to the pair of forced inviscid Burgers' equations L_u+LL_x=σ(u,x,y,L) and M_u+MM_y=σ'(u,x,y,M) where u,x,y are Bondi coordinates of scri. The functions σ and …

The present paper solves the problem of the group classification of the general Burgers' equation $u_t=f(x,u)u_x^2+g(x,u)u_{xx}$, where $f$ and $g$ are arbitrary smooth functions of the variable $x$ and $u$, by using Lie method. The paper is one of the few applications of an algebraic approach to the problem of group c…

We study the problem of coupling Einstein's equations to a relativistic and physically well-motivated version of the Navier-Stokes equations. Under a natural evolution condition for the vorticity, we prove existence and uniqueness in a suitable Gevrey class if the fluid is incompressible, where this condition is given …

Bayesian PINNs optimize loss weights for PDEs and data.

Deep learning predicts dynamics from sparse data.

In this paper, we will generalize the Bott-Virasoro group, applying the concept of the connection cochain, and derive the Euler equations corresponding to the generalized Bott-Virasoro group. We will show the relationships between the new Euler equations and the old ones. Moreover, we will study the geodesic equation c…

GrADE uses graph neural networks and Neural ODE for solving time-dependent nonlinear PDEs efficiently.

The physics informed neural network (PINN) is evolving as a viable method to solve partial differential equations. In the recent past PINNs have been successfully tested and validated to find solutions to both linear and non-linear partial differential equations (PDEs). However, the literature lacks detailed investigat…

Deep autoencoder finds linear PDE coordinates for nonlinear equations.

PDE-NetGen converts physical equations to neural networks for various scientific problems.

New method learns PDE solutions from low-fidelity data.

Uniqueness of 1D bi-Schrödinger flow proven from flat torus to compact space.

In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which are used in computer aided geometric design. We consider these curves in the framework of the similarity geometry and characterize them as invariant curves under the integrable flow on plane curves which is gov…

RS-PINN uses randomized smoothing to speed up high-dimensional PDE simulations without sacrificing accuracy.

New method uses models from regularity structures as features in machine learning.

TNet combines DL with physics models to solve inverse problems efficiently.

The paper analyzes thin-shell limits for viscous operators on Riemannian hypersurfaces.

We consider a natural Riemannian metric on the infinite dimensional manifold of all embeddings from a manifold into a Riemannian manifold, and derive its geodesic equation in the case $\Emb(\Bbb R,\Bbb R)$ which turns out to be Burgers' equation. Then we derive the geodesic equation, the curvature, and the Jacobi equat…

High-dimensional PDEs have been a longstanding computational challenge. We propose to solve high-dimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator, initial condition, and boundary conditions. Our algorithm is meshfree, which is key since mesh…

We consider some elementary aspects of the geometry of the space of probability measures endowed with Wasserstein distance. In such a setting, we discuss the various terms entering Perelman's shrinker entropy, and characterize two new monotonic functionals for the volume-normalized Ricci flow. One is obtained by a resc…

Study on the geometric Dyson Brownian motion of non-square matrix products.