The method of choice for integrating the time-dependent Fokker-Planck equation in high-dimension is to generate samples from the solution via integration of the associated stochastic differential equation. Here, we introduce an alternative scheme based on integrating an ordinary differential equation that describes the flow of probability. Unlike the stochastic dynamics, this equation deterministically pushes samples from the initial density onto samples from the solution at any later time. The method has the advantage of giving direct access to quantities that are challenging to estimate from stochastic trajectories, such as the probability current, the density itself, and its entropy. The probability flow equation depends on the gradient of the logarithm of the solution (its "score"), and so is a-priori unknown. To resolve this dependence, we model the score with a deep neural network that is learned on-the-fly by propagating a set of samples according to the instantaneous probability current. Our approach is based on recent advances in score-based diffusion for generative modeling, but the training procedure is self-contained and does not require samples from the target density to be available beforehand. To demonstrate the validity of the approach, we consider several examples from the physics of interacting particle systems; we find that the method scales well to high-dimensional systems and accurately matches available analytical solutions and moments computed via Monte-Carlo.

翻译：将基于时间的 Fokker- Planck 方程式整合到高dimenion 中的选择方法是通过整合相关随机差异方程式,从溶液中生成样本。 在这里, 我们推出一个基于整合普通差异方程式的替代方案, 描述概率的流量。 与随机动态不同, 此方程式将样本从初始密度的确定性推向从初始密度到任何以后的溶液样本的确定性。 该方程式的优点是, 直接获取从随机轨迹中估算出具有挑战性的数量, 如概率当前、 密度本身 及其英特基。 概率流方程式取决于解决方案的对数梯度( 其“ 分数 ” ), 其概率流方程式取决于解决方案的对数梯度, 因而是未知的。 为了解决这一依赖性, 我们用一个深层的神经网络来模拟评分, 通过按照瞬时概率对一组样本进行推导, 方法的优势在于从一个基于分法基础的模型的传播进展, 但是培训程序是自足的, 不需要从目标方位分析方法的样本, 的精确度方法的精确度 。, 我们从可找到一个精确的精确的精确的精确的精确的精确度, 。