# Adaptive Probability Flow Residual Minimization for High-Dimensional Fokker-Planck Equations

arXiv: https://arxiv.org/abs/2512.19196 · HTML: https://mlsanity.com/paper/2512.19196/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 62 references that are in the map.
- First posted on arXiv 22 Dec 2025.

## Builds on (most cited first, 25 of 25)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Convergence for score-based generative modeling with polynomial complexity](https://mlsanity.com/paper/2206.06227/) (2022)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [Failure-informed adaptive sampling for PINNs](https://mlsanity.com/paper/2210.00279/) (2022)
- [Statistical Efficiency of Score Matching: The View from Isoperimetry](https://mlsanity.com/paper/2210.00726/) (2022)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [Probability flow solution of the Fokker-Planck equation](https://mlsanity.com/paper/2206.04642/) (2022)
- [Adaptive deep density approximation for Fokker-Planck equations](https://mlsanity.com/paper/2103.11181/) (2021)
- [Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equations](https://mlsanity.com/paper/2402.07465/) (2024)
- [Solving time dependent Fokker-Planck equations via temporal normalizing flow](https://mlsanity.com/paper/2112.14012/) (2021)
- [Self-Consistent Velocity Matching of Probability Flows](https://mlsanity.com/paper/2301.13737/) (2023)
- [Adaptive deep density approximation for stochastic dynamical systems](https://mlsanity.com/paper/2405.02810/) (2024)
- [Weak Generative Sampler to Efficiently Sample Invariant Distribution of Stochastic Differential Equation](https://mlsanity.com/paper/2405.19256/) (2024)
- [Variational conditional normalizing flows for computing second-order mean field control problems](https://mlsanity.com/paper/2503.19580/) (2025)

## Built on by (most cited first, 1 of 1)

- [Quantum Dynamics via Score Matching on Bohmian Trajectories](https://mlsanity.com/paper/2604.25137/) (2026)

## Abstract (arXiv)

Solving high-dimensional Fokker-Planck (FP) equations remains a challenging problem in computational physics and stochastic dynamics, due to the curse of dimensionality, unbounded domains, and complex probability landscapes. In this work, we propose an adaptive probability flow residual minimization (A-PFRM) method for this problem. The second-order FP equation is reformulated as an equivalent first-order continuity equation associated with the probability flow ordinary differential equation, based on which a loss function is constructed to train neural network approximations without Hessian computation. To further improve the computational efficiency, the Hutchinson trace estimator is applied to compute the divergence in the corresponding score function, such that the training time can be dimension-independent on GPUs. Adaptive sampling strategies are employed to generate the collocation points, and our analysis shows that the Kullback-Leibler divergence between our A-PFRM approximation and the exact solution is bounded by the residual loss weighted by the estimated density function. Numerical experiments are presented to demonstrate the performance of A-PFRM, which include Ornstein-Uhlenbeck (OU) processes problems, Brownian motions with time-varying diffusion, and Geometric OU processes featuring non-Gaussian solutions up to one hundred dimensions.