Entropy-dissipation Informed Neural Network for McKean-Vlasov Type PDEs

We extend the concept of self-consistency for the Fokker-Planck equation (FPE) to the more general McKean-Vlasov equation (MVE). While FPE describes the macroscopic behavior of particles under drift and diffusion, MVE accounts for the additional inter-particle interactions, which are often highly singular in physical systems. Two important examples considered in this paper are the MVE with Coulomb interactions and the vorticity formulation of the 2D Navier-Stokes equation. We show that a generalized self-consistency potential controls the KL-divergence between a hypothesis solution to the ground truth, through entropy dissipation. Built on this result, we propose to solve the MVEs by minimizing this potential function, while utilizing the neural networks for function approximation. We validate the empirical performance of our approach by comparing with state-of-the-art NN-based PDE solvers on several example problems.Comment: Accepted to NeurIPS 202

Mean-field limit of Non-exchangeable interacting diffusions with singular kernels

The mean-field limit of interacting diffusions without exchangeability, caused by weighted interactions and non-i.i.d. initial values, are investigated. The weights could be signed and unbounded. The result applies to a large class of singular kernels including the Biot-Savart law. We demonstrate a flexible type of mean-field convergence, in contrast to the typical convergence of $\frac{1}{N}\sum_{i=1}^N\delta_{X_i}$. More specifically, the sequence of signed empirical measure processes with arbitrary uniform $l^r$-weights, $r>1$, weakly converges to a coupled PDE's, such as the dynamics describing the passive scalar advected by the 2D Navier-Stokes equation. Our method is based on a tightness/compactness argument and makes use of the systems' uniform Fisher information. The main difficulty is to determine how to propagate the regularity properties of the limits of empirical measures in the absence of the DeFinetti-Hewitt-Savage theorem for the non-exchangeable case. To this end, a sequence of random measures, which merges weakly with a sequence of weighted empirical measures and has uniform Sobolev regularity, is constructed through the disintegration of the joint laws of particles.Comment: 34 page

Mean field limit for stochastic particle systems with singular forces

In this thesis, we systematically study the mean field limit for large systems of particles interacting through rough or singular kernels by developing a new statistical framework, based on controlling the relative entropy between the $N-$particle distribution and the limit law through identifying new Laws of Large Numbers. We study both the canonical 2nd order Newton dynamics and the 1st order (kinematic) systems, leading to McKean-Vlasov systems in the large $N$ limit. For the 2nd order case, we only require that the interactions $K$ be bounded. The control of the relative entropy implies the mean field limit and the propagation of chaos through the strong convergence of all the marginals. For the 1st order case, with the help from noise we can even obtain the mean field limit for interactions $K \in W^{-1, \infty}$, {\em i.e.} the anti-derivatives of $K$ are bounded (or even unbounded with weak singularity). To our knowledge, this is the first time the relative entropy method applied to obtain the mean field limit. Compared to the classical framework with $K \in W^{1, \infty}$, our results show another critical scale $K \in L^\infty$ for the mean field limit. Our results are quantitative: we can provide precise control of the relative entropy and hence the convergence of the marginals. We expect that the relative entropy method will be another standard tool in the study of the mean field limit. This thesis resulted in the publications \cite{JW1, JW2, JW3}

Winding number, density of states and acceleration

Winding number and density of states are two fundamental physical quantities for non-self-adjoint quasi-periodic Schr\"odinger operators, which reflect the asymptotic distribution of zeros of the characteristic determinants of the truncated operators under Dirichlet boundary condition, with respect to complexified phase and the energy respectively. We will prove that the winding number is in fact Avila's acceleration and it is also closely related to the density of states by a generalized Thouless formula for non-self-adjoint Schr\"odinger operators and Avila's global theory

On Mean Field Limit and Quantitative Estimates with a Large Class of Singular Kernels: Application to the Patlak-Keller-Segel Model

In this note, we propose a new relative entropy combination of the methods developed by P.--E. Jabin and Z.~Wang [Inventiones (2018)] and by S. Serfaty [Proc. Int. Cong. of Math, (2018) and references therein] to treat more general kernels in mean field limit theory. This new relative entropy may be understood as introducing appropriate weights in the relative entropy developed by P.-E. Jabin and Z. Wang (in the spirit of what has been recently developed by D.~Bresch and P.--E. Jabin [Annals of Maths (2018)]) to cancel the more singular terms involving the divergence of the flow. As an example, a full rigorous derivation (with quantitative estimates) of the Patlak-Keller-Segel model in some subcritical regimes is obtained. Our new relative entropy allows to treat singular potentials which combine large smooth part, small attractive singular part and large repulsive singular part