Fokker--Planck Equation

We study the long-time behavior of Wasserstein gradient flows for interaction energies $$\mathsf E[μ] = \frac12\iint_{M\times M}K(x,y)\,\mathrm dμ(x)\,\mathrm dμ(y)$$ on a closed manifold $M$. For kernels diagonal in a Laplace eigenbasis with nonnegative spectral coefficients, we prove a differential inequality relating the relative entropy to the energy gap. Consequently, for any nonnegative initial density $u_0\in L^p(M)$, $p>1$, the energy gap is integrable in time and satisfies $$\mathsf E[μ_t]-\mathsf E_{\min}=o(t^{-1}).$$ If all spectral coefficients are positive, the flow converges weakly to the constant measure. These interaction energies need not be geodesically convex in Wasserstein space, and the associated flows contain no diffusion; their global convergence therefore does not follow from standard Wasserstein gradient flow theory. The kernels covered by our results include zonal kernels on spheres, kernels arising in transformer models, regularized Riesz kernels, and inverse fractional Laplacian kernels. We also investigate the sharpness of the $o(t^{-1})$ rate. For any smooth kernel in this class with infinitely many positive spectral coefficients and any $δ>0$, we construct a solution of the linearized flow whose energy is comparable to $t^{-1-δ}$ along a sequence of times tending to infinity. Moreover, for any $δ>0$, by choosing a suitable inverse fractional Laplacian kernel on the flat torus, we construct an exact solution of the nonlinear Wasserstein gradient flow whose energy is comparable to $t^{-1-δ}$. The nonlinear construction is based on uniform-in-time estimates for the evolution of the dyadic Fourier coefficient blocks and a blockwise energy-persistence argument. These estimates also yield a uniform-in-time quantitative comparison between the nonlinear Wasserstein gradient flow and its linearization.

The local dark matter density determines the strength of the signal expected in direct-detection experiments, yet published estimates from stellar motions disagree by more than their errors, and the most recent machine-learning analysis of Gaia data finds a local density consistent with zero. According to Jeans' theorem, a distribution function built from integrals of motion satisfies the collisionless Boltzmann equation (CBE) trivially for any choice of potential, so a search that simultaneously fits the distribution function and the potential to the CBE identifies neither. Our pipeline instead estimates the distribution function in isolation, linearizing the equation in terms of accelerations and allowing for direct measurement of the local force field, and then fits closed forms to that field via symbolic regression. Throughout, we find that the usable information lies not in the CBE residual but in the stellar number counts, the observable most distorted by survey selection. Along the vertical profile, our recovered potential agrees with the classical self-gravitating isothermal disc.

A novel advective Fisher-Rao metric is introduced for optimization tasks on paths of probability measures governed by the continuity equation. This metric is shown to lead to optimal descent directions. It is then shown that this metric arises naturally from three different perspectives: As the rescaled zero-noise limit of the Fisher-Rao metric on path measures, as the expected value of the second variation of the Freidlin--Wentzell large deviation rate functional, and as the Hessian of the Benamou--Brenier action functional from dynamic optimal transport. We supplement this geometric construction with computational experiments. Here, we demonstrate empirically that the advective Fisher-Rao metric yields the desired optimal fitting of probability densities, whereas the Gauss--Newton method yields optimal fitting of velocity fields.

We introduce conditional cylindrical neural networks for approximating functionals of conditional laws in McKean-Vlasov equations with common noise. Fourier moments of the initial law and truncated signatures of the time augmented common noise are mapped by a mixture density network to a Gaussian mixture approximation of the conditional law. A cylindrical neural network then evaluates the target functional through analytic integrals against this predicted measure. Rough path well posedness and stability provide a conditional law map that is continuous in the initial distribution and the rough driver and agrees almost surely with the classical conditional law at the Itô Brownian lift. Combining this continuity with Fourier separation, signature uniqueness, Wasserstein density of Gaussian mixtures, and neural universal approximation, we prove an $L^2$ universal approximation theorem for continuous square integrable functionals. The numerical study implements the resulting two stage procedure on six examples, including non Gaussian initial laws, nonlinear drift, multiplicative common noise, and a two dimensional state. Independent particle references are used when no closed form law is available. The learned conditional law and functional approximations consistently improve on the empirical particle plug in, and additional experiments examine feature sensitivity, training from one terminal observation per common noise scenario, and Itô--Stratonovich consistency.

Modal synthesis is a widely-used technique for simulation of musical instrument dynamics. In the linear case, a modal decomposition leads to an uncoupled system of damped and forced harmonic oscillators which can be efficiently solved by standard time-stepping methods. However, extensions to nonlinear problems are challenging due to the presence of products of modal expansions in the governing equations. In the case of the Föppl-von Kármán plate, the nonlinear coupling between the modes is described by a fourth-order tensor and is prohibitively expensive to evaluate in the modal domain. In this work, we propose a pseudospectral method in which the products are evaluated on a grid in the spatial domain while spatial derivatives are computed exactly in the modal domain. Discrete sine and cosine transforms between the modal and spatial domains are used to impose simply supported boundary conditions for the plate. Finally, we prove non-negativity of the nonlinear potential energy of the system and employ a scalar auxiliary variable technique for explicit and stable time integration in the modal domain. As a result, we reduce the computational cost of modal synthesis while preserving its advantages like a precise control over the simulated frequency range. Sound examples are presented.

Motivated by marginal distribution flows of reflected diffusions in bounded domains, we investigate when a density/flux pair solving a no-flux continuity equation admits a regular Lagrangian flow that remains in the closed domain and generates the prescribed density flow. We give sufficient conditions in terms of interior bounded-variation regularity, bounded-variation control on a boundary collar, a one-sided bound on an absolutely continuous divergence, and vanishing normal trace of the velocity. The proof uses the fact that tangency removes the singular boundary contribution to the divergence of the zero extension, thereby making the extended velocity admissible for the Ambrosio-DiPerna-Lions theory. We show that these boundary assumptions cannot be jointly relaxed so as to admit a boundary current mechanism. We construct an explicit smooth density/flux pair carrying a boundary current. Its density evolution is unique in a weighted class and its characteristics are unique, confined and transport the marginals, yet it admits no regular Lagrangian flow because the compressibility bound fails arbitrarily close to the initial time. We also establish two uniqueness results for no-flux Fokker-Planck equations: a duality result for bounded measurable drifts and a weighted energy result for entrance-type drifts singular at the boundary. Our results provide a rigorous mathematical justification for using the ODE-based sampling of reflected diffusion models under minimal regularity assumptions on the coefficients, and also indicate when such ODE-based samplers may fail.

Finite-time driving of stochastic systems generates excess dissipation, causing the evolving probability distribution to lag behind the instantaneous equilibrium, and consequently degrading the convergence of nonequilibrium free energy estimators based on the Jarzynski equality. Escorted free energy simulations address the non-adiabatic lag by engineering control fields $\mathbf{u}$ that eliminate the lag, enforcing the trajectory-wise equality $\mathcal{W}_\mathbf{u} = Δ\mathcal{F}$, and yielding zero-variance estimators. However, constructing the escorting field in closed form remains a challenge, approached variously through flow-field methods, targeted free energy perturbation, or learned diffeomorphisms. In this work, we construct a complementary numerical framework based on gauge-type transforms instead of generalized coordinate transforms for perfect escorting based on the exact spectral decomposition of the time-dependent Fokker-Planck generator. The biorthogonal decomposition of the Liouville operator directly yields a counterdiabatic correction whose action on the instantaneous equilibrium distribution exactly cancels the non-adiabatic lag at arbitrary driving speed in formal analogy with shortcuts-to-adiabaticity techniques such as Berry's transitionless driving for quantum systems. Numerical verification for simulations of an overdamped particle in a time-varying double-well potential and harmonic traps confirms that the counterdiabatic condition is satisfied to machine precision, with the non-adiabatic lag suppressed by roughly twelve orders of magnitude in total variation distance and sixteen orders in KL divergence relative to the unescorted dynamics. As a diagnostic, we demonstrate vanishing dissipated work $\mathcal{W}_{\text{diss}}(t) \approx 0$ for the deterministically propagated Fokker-Planck density across all protocol speeds.

Randomized features provide a scalable approximation to kernel machines, but their performance depends strongly on the choice of feature distribution. We propose a particle-based method that learns this distribution by optimizing kernel-target alignment while regularizing particles with a Riesz/Coulomb repulsive potential. The resulting Hamiltonian yields diverse, task-adaptive random features and admits a mean-field description through a McKean--Vlasov equation. We instantiate the method in linearized Transformer attention by learning positive random-feature maps in a first alignment phase, then freezing the kernel and training the remaining network parameters with cross-entropy. Experiments on synthetic classification and sentence-level benchmarks show that learned kernelized attention can improve accuracy, calibration, and robustness for several feature maps while preserving linear-attention inference complexity.

This paper develops a model-free reinforcement learning framework for continuous--time extended mean field control problems, where both the dynamics and reward may depend on the joint distribution of states and controls. We adopt deterministic feedback policies, under which the state--action distribution is induced directly as a push--forward of the state law. This avoids optimization over stochastic kernels and bypasses key limitations of existing approaches in extended mean field settings. We first establish a model--free sensitivity formula for parameterized McKean--Vlasov dynamics and use it to derive a deterministic policy gradient formula expressed through an advantage--rate function on the Wasserstein space. We then refine this formula by introducing local value and advantage--rate representations that depend on the state, action, and joint state--action distribution, yielding a policy gradient that includes both action derivatives and measure--derivative terms with respect to the control distribution. These characterizations lead to a martingale--based learning principle and motivate a continuous--time deep deterministic policy gradient algorithm combining particle approximations, measure--dependent neural networks, temporal--difference learning, and exploration in either action or parameter space. Numerical experiments on stochastic Cucker--Smale consensus control and optimal liquidation with trade crowding demonstrate the efficiency, stability, and robustness of the proposed method, including problems with explicit dependence on the control distribution.

We formalize a research result in the Lean 4 proof assistant by having a mathematician direct an AI system, and frame the activity as a formalization game. The objective is to turn a LaTeX document into Lean. The game is won when the development compiles, contains no sorry, and a machine check shows the target theorems rest on Lean's foundational axioms alone. Reuse is a second check, by a definition we introduce: whether the development yields a self-contained layer of general mathematics the wider library could absorb. The case study is a complete, axiom-clean formalization of well-posedness for the nonlinear Vlasov equation via Dobrushin's mean-field route -- existence, uniqueness, the stability estimate and mean-field limit, and a short-window superposition principle (weak solutions are Lagrangian). The human's role was to direct, not to write proofs: to scope the definitions, steer the decompositions, and triage the library's gaps; the AI agent executed. The formalization certifies the proof of each statement as written; whether the written statement is the intended theorem stays the mathematician's judgment. The optimal-transport machinery that fell out of the build (in particular, properties of the Wasserstein-1 metric and the Kantorovich-Rubinstein duality theorem) separates into a self-contained layer that compiles against Mathlib alone: about a sixth of the development (49 of 299 declarations), behind a 22-declaration interface with no reverse dependency. The headline theorems ran in about a week, the full development in about a month. We report the quantitative claims as observations of one game, not as general laws. The game's rules name no particular system, so the methodological framing is meant to outlast the tools of any one run.

The use of ordinary and stochastic differential equations has led to substantial progress in generative machine learning with applications to, for example, image, video and biomolecule generation. This paper provides a self-contained and informal introduction to the differential equations, the probabilistic framework for using them in generative modeling and the Fokker--Planck equation that governs the temporal evolution of the marginal distribution of the stochastic variables of the differential equations. The variational lower bound on the log-likelihood (the evidence lower bound, ELBO) is derived and used as a general starting point for a discussion of diffusion models, score matching, and flow matching. All of these approaches may be viewed as specific parameterizations of the most general variational approach. A one-dimensional density modeling problem is used as a simple example to compare different parameterizations.

Modeling chaotic systems is crucial yet challenging. Inverse problems in chaotic dynamics, namely inferring initial conditions from final states, remain largely unsolved because of ill-posedness, non-uniqueness, instability, and potentially chaotic time-reverse dynamics. We address this open problem with Bidirectional Conditional Flow Matching (Bi-CFM), which learns bidirectional mappings between distributions of initial and final states to capture the stochasticity of chaotic evolution and mitigate exponential error accumulation over time. Furthermore, for systems with conservation laws, we extend it to Conservation-constrained Bi-CFM (CBi-CFM). Across the classic Lorenz, Circuit, and high-dimensional Lorenz 96 systems, Bi-CFM improves five distribution-level metrics over baselines while achieving a speedup of more than two orders of magnitude. In the three-body planet-planet scattering problem in planetary dynamics, CBi-CFM better respects conservation laws, with conservation errors comparable to those of the ground truth. Finally, on real observations of globular clusters, collisional million-body systems shaped by $\sim 10^{10}$ years (10 Gyr) of evolution, our method represents an advance in accuracy, establishing a scalable route to solving inverse problems of long-timescale real-world chaotic dynamics.

The space $\mathcal{P}_2(\mathbb{R}^d$) of probability measures with finite second moment carries a natural geometry: the quadratic Wasserstein distance W_2 makes it a complete metric space and, following Otto, a (formal) Riemannian manifold whose geodesics are the optimal-transport interpolations. On this manifold, the gradient flow of the free energy F(rho) = KL(rho || π) is exactly the Fokker-Planck equation, and its implicit-Euler discretization is the JKO scheme. This is the geometry underlying diffusion models: the forward process descends the free energy, and each denoising step realizes one JKO step, which recovers DDPM, DDIM, NCSN/SMLD, and Energy Matching; this is one scheme, not separate theories. The same manifold supports a second variational principle. Its geodesics - the minimum-action curves of the Benamou-Brenier formula - are precisely the optimal-transport paths that Flow Matching learns. Fixing both endpoints and following the geodesic, generation becomes a deterministic ODE along a straight line, hence far fewer sampling steps. Placing both families of models on one manifold makes their relationship exact: diffusion follows a free-energy gradient flow, an initial-value problem; optimal-transport Flow Matching follows a Wasserstein geodesic, a boundary-value problem. The two reach the same endpoints along different paths.

The Fokker-Planck equation (FPE) plays a pivotal role in describing the time evolution of probability density functions (PDFs) for systems governed by stochastic dynamics. In this work, we propose a conditional normalizing flow-based physics-informed neural network (PINN) framework for efficiently approximating the solution operator of the FPE for a whole range of initial conditions. Leveraging the Chapman-Kolmogorov equation for Markovian stochastic processes, the problem is reformulated into approximating a transition PDF starting at initial time from a Dirac mass centered at an arbitrary point. The PDF of an associated linearized stochastic differential equation (SDE) is employed as the base distribution for the normalizing flow, providing a good approximation of the target PDF, especially for small times, and thereby avoiding the singularity of the map associated with the Dirac delta initial distribution. Furthermore, a time-weighted loss function is introduced to mitigate numerical instabilities arising at small times, achieving a balance between causality and training difficulty as time progresses. A variety of numerical experiments are presented to illustrate the effectiveness and robustness of the proposed method.

Score-based diffusion models are typically trained by minimizing the $L^2$ score matching error, and standard theoretical analyses rely on this quantity to bound the sampling discrepancy between the learned and target distributions. We show the $L^2$ score error is not the right intrinsic measure of marginal distributional quality: a learned diffusion model can incur arbitrarily large $L^2$ score error while perfectly matching the target distribution. By decomposing score errors into a gradient and a solenoidal component (a Helmholtz-Hodge decomposition), we identify the geometric reason behind this: only the gradient component enters the marginal Fokker-Planck dynamics, while the solenoidal component is structurally invisible. We make this precise in three results. First, building on the corrected geometry, we prove an impossibility result: no monotone function of the $L^2$ score error can uniformly lower bound any divergence between the learned and target distributions. Second, we derive an upper bound on the Kullback-Leibler divergence that depends only on the observable gradient component of the error, tightening the standard Girsanov bound for generic score networks, and identifying its looseness as the cost of operating on path-space rather than marginal-space dynamics. Third, we give a tractable estimator of the gradient component via a dual Sobolev identity, which is shown to empirically correlate substantially better with sample quality than the full $L^2$ error.

One of the primary challenges in Bayesian inference on the parameters of a diffusion model from discrete observations is the unavailability of an analytical expression for the transition density function between consecutive observation times, which is needed to derive the likelihood function. Extending previous studies that solve Fokker-Planck (FP) type partial differential equations with Normalizing Flows, we propose a new Normalizing Flow architecture to learn the transition density function of the diffusion process between two observation times. We do so by solving in a Neural Galerkin framework the associated FP equation with a Dirac mass as initial condition, over a specified training distribution of the initial datum and the coefficients of the diffusion. We specifically focus on processes whose diffusion matrix vanishes in certain inaccessible boundary regions, such as Stochastic Volatility models that satisfy a Feller condition. The product of the obtained transition densities evaluated along the observed trajectory approximates the likelihood function, thereby enabling cheap posterior sampling via Markov chain Monte Carlo (MCMC). After the offline training phase, inference becomes significantly more efficient, as it avoids the need to solve the FP equation in real time for each parameter proposed by the MCMC sampler or to rely on other likelihood-free methods for Bayesian inference that involve repeated simulation of diffusion bridges.

Several important learning tasks can be formulated as minimizing an entropy-regularized objective over an appropriate space of probability distributions. Mean-field Langevin dynamics (MFLD) facilitate computation in this general context, casting the minimizer as the invariant distribution of a McKean--Vlasov process, which can be numerically discretized using $N$ particles and thus simulated. However, simulating this interacting particle system has computational complexity of order $N^2$. Motivated by recent research into \emph{kernel thinning}, we propose \texttt{KT-MFLD}, in which each particle interacts only with a thinned particle coreset of size $\mathcal{O}(N^{\frac{1}{2}})$. \texttt{KT-MFLD} thus reduces the computational complexity to order $N^{\frac{3}{2}}$ while, under mild regularity conditions, achieving the same convergence guarantees (up to logarithmic factors) as MFLD. Our theoretical analysis is empirically confirmed on tasks including the training of student-teacher neural networks, quantization with maximum mean discrepancy, and computation of predictively-oriented posteriors in a post-Bayesian framework.

Stochastic Gradient Descent (SGD) is commonly modeled as a Langevin process, assuming that minibatch noise acts as Brownian motion. However, this approximation relies on a continuous-time limit and a sqrt(eta) noise scaling that does not match the discrete SGD update at finite learning rate. In this work, we propose an alternative formulation of SGD as deterministic dynamics in a fluctuating loss landscape induced by minibatch sampling. Starting directly from the discrete update, we derive a master equation for the parameter distribution and obtain a discrete Fokker--Planck equation that differs from the standard Langevin form at order eta^2. Using this framework, we analyze SGD dynamics near critical points of the loss. We show that the behavior decomposes along the eigenbasis of the mean Hessian into qualitatively distinct regimes. In particular, nearly-flat directions do not admit a stationary distribution: the variance grows over time, corresponding to effective diffusion along valleys with a coefficient proportional to the learning rate. We provide empirical evidence supporting these predictions on neural network models in computer vision and natural language processing, observing a clear qualitative separation between confined and diffusive modes.

Reconstructing continuous physical fields from sparse measurements is a central inverse problem, but data-driven generative models can produce states that violate governing dynamics. We introduce a physics-informed generative solver that separates stable prior learning from inference-time enforcement of conservation laws. Martingale-Regularized Score Matching regularizes score pretraining with a Score Fokker-Planck constraint, yielding a dynamically stable prior. Physics-Informed Implicit Score Sampling then guides denoising trajectories by gradients of physical residuals, projecting samples toward admissible manifolds without retraining. In acoustics, the method co-generates pressure and particle velocity from sparse sensors, enabling dense virtual arrays that suppress spatial aliasing. The same framework generalizes to real-world ERA5 meteorological fields under extreme sparsity. Together, this work establishes a rigorous and generalizable paradigm for solving high-dimensional inverse problems, bridging the gap between generative artificial intelligence and first-principles science.

We introduce \textbf{QuantFPFlow}, a reinforcement learning framework that integrates quantum amplitude estimation into the Fokker--Planck~(FP) formulation of stochastic policy optimisation. Classical continuous-space RL agents must estimate the FP partition function $Z = \int e^{-V(\mathbf{x})/D}\,d\mathbf{x}$ at cost $\calO(1/\varepsilon^{2})$; QuantFPFlow replaces this with a Grover-amplified amplitude estimator achieving $\calO(1/\varepsilon)$ -- a provable quadratic speedup. While the full quantum acceleration requires fault-tolerant hardware, the quantum-inspired classical simulation demonstrated here already exhibits the $\calO(1/\varepsilon)$ algorithmic structure. The estimated stationary distribution $\rhostar$ drives a theoretically grounded exploration bonus $\Raug = \Renv + α\log(1/\rhostar(s))$. This bonus steers the agent toward globally optimal regions of multimodal reward landscapes while simultaneously constraining policy variance through FP diffusion matching. On a continuous-control task specifically designed to expose local-optima failure, QuantFPFlow achieves mean reward $1{,}295.7 \pm 423.2$ versus $1{,}284.0 \pm 474.0$ for Soft Actor-Critic~(SAC), while discovering the global optimum \textbf{10.4,% more frequently} (33.9,% vs.\ 30.7,%). Policy entropy remains near $H(π)\approx 6.5$,nats throughout training, whereas SAC collapses to $1.5$,nats, confirming that FP diffusion matching actively prevents premature convergence. Dimensionality experiments further show computational scaling of $\calO(d^{0.35})$ for QuantFPFlow versus $\calO(d^{0.76})$ for classical FP estimation.

Learning the underlying potential energy of stochastic gradient systems from partial and noisy observations is a fundamental problem arising in physics, chemistry, and data-driven modeling. Classical approaches often rely on direct regression of governing equations or velocity fields, which can be sensitive to noise and external perturbations and may fail when observations are incomplete. In this work, we propose a structure-aware, energy-based learning framework for inferring unknown potential functions in generalized diffusion processes, grounded in the energetic variational approach. Starting from the energy-dissipation law associated with the Fokker-Planck equation, we construct loss functions based on the De Giorgi dissipation functional, which consistently couple the free energy and the dissipation mechanism of the system. This formulation avoids explicit enforcement of the governing partial differential equation and preserves the underlying variational structure of the dynamics. Through numerical experiments in one, two, and three dimensions, we demonstrate that the proposed energy-based loss exhibits enhanced robustness with respect to observation time, noise level, and the diversity and amount of available training data. These results highlight the effectiveness of energy-dissipation principles as a reliable foundation for learning stochastic diffusion dynamics from data.

Recent work has shown that diffusion models trained with the denoising score matching (DSM) objective often violate the Fokker--Planck (FP) equation that governs the evolution of the true data density. Directly penalizing these deviations in the objective function reduces their magnitude but introduces a significant computational overhead. It is also observed that enforcing strict adherence to the FP equation does not necessarily lead to improvements in the quality of the generated samples, as often the best results are obtained with weaker FP regularization. In this paper, we investigate whether simpler penalty terms can provide similar benefits. We empirically analyze several lightweight regularizers, study their effect on FP residuals and generation quality, and show that the benefits of FP regularization are available at substantially lower computational cost. Our code is available at https://github.com/OnnoNiemann/fp_diffusion_analysis.

We propose a new Neural Galerkin Normalizing Flow framework to approximate the transition probability density function of a diffusion process by solving the corresponding Fokker-Planck equation with an atomic initial distribution, parametrically with respect to the location of the initial mass. By using Normalizing Flows, we look for the solution as a transformation of the transition probability density function of a reference stochastic process, ensuring that our approximation is structure-preserving and automatically satisfies positivity and mass conservation constraints. By extending Neural Galerkin schemes to the context of Normalizing Flows, we derive a system of ODEs for the time evolution of the Normalizing Flow's parameters. Adaptive sampling routines are used to evaluate the Fokker-Planck residual in meaningful locations, which is of vital importance to address high-dimensional PDEs. Numerical results show that this strategy captures key features of the true solution and enforces the causal relationship between the initial datum and the density function at subsequent times. After completing an offline training phase, online evaluation becomes significantly more cost-effective than solving the PDE from scratch. The proposed method serves as a promising surrogate model, which could be deployed in many-query problems associated with stochastic differential equations, like Bayesian inference, simulation, and diffusion bridge generation.

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.