We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. Leveraging stochastic representations and Walk-on-Spheres methods, we develop novel losses for neural networks based...
It is known that there is a one-to-one mapping between oriented directed graphs and zero-sum replicator dynamics (Lotka-Volterra equations) and that furthermore these dynamics are Hamiltonian in an appropriately defined nonlinear Poisson bracket. In...
The variation problem associated with the solution of Dirac's relativistic electron equation is reviewed here. Derivation of the min-max theorem is discussed. A new observation is that the spurious roots of negative energy satisfy a max-min theorem....
A unique analytic solution of the generalized Dixon nonhomogeneous integral equation is derived in $C^{1}[0,A],\ A > 0$. It is written in terms of the Neumann series, which is expressed as a double series of residues at multiple poles of powers of th...
The valence Fock-state wavefunctions of the light-front QCD Hamiltonian satisfy a relativistic equation of motion with an effective confining potential $U$ which systematically incorporates the effects of higher quark and gluon Fock states. If one re...
We have shown in a recent collaboration that the Cauchy problem for the multi-dimensional Burgers equation is well-posed when the initial data u(0) is taken in the Lebesgue space L 1 (R n), and more generally in L p (R n). We investigate here the sit...
A geometric framework for describing and solving time-dependent implicit differential equations F(t,x,x')=0 is studied, paying special attention to the linearly singular case, where F is affine in the velocities: A(t,x)x' = b(t,x). This framework i...
Lagrangian systems with nonholonomic constraints may be considered as singular differential equations defined by some constraints and some multipliers. The geometry, solutions, symmetries and constants of motion of such equations are studied within...
This work investigates the boundary stabilization of flows in star-shaped and tree-shaped networks of open channels governed by the Saint-Venant equations with a friction term. Due to the existence of the friction term, the steady-states are non-unif...
We systematically study a numerical procedure that reveals the asymptotically self-similar dynamics of solutions of partial differential equations (PDEs). This procedure, based on the renormalization group (RG) theory for PDEs, appeared initially in...
We provide a conjecture for the following two quantities related with the spin-$\frac{1}{2}$ isotropic Heisenberg model defined over rings of even lengths: (i) the number of the solutions to the Bethe ansatz equations which correspond to non-zero Bet...
We study the asymptotic behavior of low-lying eigenvalues of spatially cut-off $P(φ)_2$-Hamiltonian under semi-classical limit. The corresponding classical equation of the $P(φ)_2$-field is a nonlinear Klein-Gordon equation which is an infinite dim...
In this paper, we show the increasing stability of the inverse source problems for the acoustic wave equation in the full space R3.The goal is to understand increasing stability for wave equation in the time domain. If the time and spatial variables...
In this work we study the optimality of increasing stability of the inverse boundary value problem (IBVP) for Schrödinger equation. The rigorous justification of increasing stability for the IBVP for Schrödinger equation were established by Isakov...
Tags: algorithm, matlab, numerical-methods, finite-difference | Score: 2
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The usual derivation of the Fokker-Planck partial differential eqn. assumes the Chapman-Kolmogorov equation for a Markov process. Starting instead with an Ito stochastic differential equation we argue that finitely many states of memory are allowed...
We construct analytic solutions to the generalized Grad-Shafranov equation, which incorporates both toroidal and poloidal flows. This is achieved by adopting a general linearizing ansatz for the free-function terms of the equation and expanding the g...
In this paper we prove the global in time well-posedness of the following non-local diffusion equation with $α\in[0,2/3)$: $$ \partial_t u = {(-\triangle)^{-1}u} \triangle u + αu^2, \quad u(t=0) = u_0. $$ The initial condition $u_0$ is positive, ra...
We study the entropy production of the sandwiched Rényi divergence under the primitive Lindblad equation with GNS-detailed balance. We prove that the Lindblad equation can be identified as the gradient flow of the sandwiched Rényi divergence of any...