Operator learning for solving Fokker-Planck equations with various initial conditions
Li Zeng, Xiaoliang Wan, Yaobin Wang, Fabio Nobile, Tao Zhou
Abstract
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.
AI Impact Assessments
(1 models)Scientific Impact Assessment
Core Contribution
This paper addresses the problem of solving Fokker-Planck equations (FPEs) efficiently across a range of initial conditions — essentially learning the solution operator rather than solving individual instances. The key insight is to reformulate the problem via the Chapman-Kolmogorov equation: instead of learning a map from initial distributions to solutions directly, the authors learn the transition PDF p(x,t|x₀), from which solutions for arbitrary initial conditions can be reconstructed by integration.
The framework has four main technical components:
1. Reformulation via transition PDF: Reducing operator learning to approximating a single transition kernel conditioned on initial state x₀ and time t.
2. Linearized SDE as base distribution: Rather than using a fixed standard normal, they use the PDF of a linearized SDE (first-order Taylor expansion of drift, zero-order of diffusion) as the normalizing flow's base distribution. This provides a time- and x₀-dependent base that closely approximates the target for small t.
3. Time-weighted loss function: Weights of t^{d/2+2} or t^{d+2} counteract the blow-up of the PINN residual near t=0 where the transition PDF approaches a Dirac delta.
4. Importance sampling for integration: A mixture proposal combining a reversed linearized SDE Gaussian and the initial distribution for evaluating the Chapman-Kolmogorov integral.
Methodological Rigor
The paper is mathematically rigorous in its theoretical contributions. Proposition 3.1 establishes that the total variation distance between the linearized and true transition PDFs is O(t) (or O(t^{3/2}) with bounded Hessian), while the KL divergence is O(t²) or O(t³). This formally justifies the base distribution choice. Proposition 3.3 proves convergence of KR maps to identity under total variation convergence — motivating the architecture design where the flow is identity at t=0. Proposition 4.1 rigorously characterizes the residual blow-up scaling, directly motivating the time-weighting strategy.
The proofs in the appendices are detailed and appear correct, drawing on established results from optimal transport theory and analysis. The connection between the Girsanov-type bound (Theorem 1.1 of Bogachev et al.) and the relative entropy estimate is cleanly applied.
However, there are some limitations in rigor:
Potential Impact
Direct applications: The method is relevant for ensemble forecasting, data assimilation, and Bayesian inference where the same stochastic dynamics are solved repeatedly with different initial conditions. This is a genuine practical need.
Methodological influence: The idea of using a problem-dependent, time-varying base distribution for normalizing flows (rather than a fixed Gaussian) is broadly applicable beyond FPEs. The analysis of residual scaling near singular initial conditions and the corresponding time-weighting strategy could transfer to other evolution PDE problems with singular data.
Limitations on impact: The restriction to bounded initial-condition domains Ω₀ and the need for the Monte Carlo integration step (which introduces its own errors) may limit practical applicability. The method also requires the drift to be smooth for the linearization, which excludes important cases (e.g., discontinuous coefficients, some mean-field models).
Timeliness & Relevance
The paper sits at the intersection of two active research areas: operator learning (DeepONet, FNO, etc.) and generative models for PDEs. The specific gap it addresses — handling varying initial conditions for FPEs without retraining — is timely and practically motivated. The concurrent work [31] on a similar approach (conditional normalizing flow for transition PDFs in a Neural Galerkin framework) confirms this is an active research direction.
The growing interest in physics-informed generative models makes this contribution relevant, though it represents an incremental rather than paradigm-shifting advance.
Strengths
1. Theoretically grounded design choices: Each architectural and algorithmic decision (base distribution, time weighting, identity initialization) is backed by formal analysis, not just heuristics.
2. Elegant problem reformulation: The Chapman-Kolmogorov decomposition cleanly separates the operator learning problem into transition PDF learning + integration.
3. Natural PDF constraints: The normalizing flow automatically satisfies positivity and unit integration, avoiding the penalty-based enforcement common in other approaches.
4. Robustness to discontinuous initial conditions: The method handles discontinuous initial distributions (e.g., uniform on bounded domains) gracefully, unlike finite difference methods.
5. Comprehensive experimental validation: Multiple SDEs (linear, nonlinear drift, state-dependent diffusion), multiple dimensions, and multiple initial conditions are tested.
Limitations
1. Scalability concerns: Testing only up to d=4 with relatively simple architectures leaves open questions about higher-dimensional applicability. The solution scaling factor of 1000 for d=4 to avoid underflow is concerning.
2. No comparison with competing operator learning methods: The paper lacks direct comparison with DeepONet, FNO, or other operator learning approaches adapted for FPEs.
3. Monte Carlo bottleneck: The final integration step via importance sampling introduces statistical errors that may dominate in practice, particularly for complex initial distributions or large t.
4. Bounded domain for x₀: The requirement x₀ ∈ Ω₀ with Ω₀ bounded limits generality.
5. Constant or simple diffusion coefficients dominate experiments: Only one example has state-dependent diffusion, and it is relatively mild.
Overall Assessment
This is a technically solid paper that makes a well-motivated contribution to an important problem. The theoretical analysis is its strongest asset, providing rigorous justification for the proposed design choices. The experimental validation is adequate but not exceptional. The main limitation is the unclear scalability to higher dimensions and more complex problems, and the absence of comparisons with alternative operator learning methods.
Generated Jun 9, 2026
Comparison History (22)
Paper 2 presents a novel scientific machine learning framework for solving the Fokker-Planck equation, which governs stochastic processes across physics, chemistry, biology, and finance. Its use of physics-informed neural networks (PINNs) and normalizing flows to handle arbitrary initial conditions offers significant real-world applicability and broad interdisciplinary impact. While Paper 1 provides strong theoretical contributions to learning general ReLUs, Paper 2's methodological innovation in solving a fundamental equation with broad practical implications gives it a higher potential for immediate and widespread scientific impact.
Paper 2 has higher likely scientific impact due to strong real-world relevance (federated learning over lossy wireless networks), broad applicability across ML, networking, and distributed systems, and a clear problem-solution framing with theoretical guarantees plus extensive experimental validation. Its methods (inverse probability weighting + AoI weighting with online channel estimation) are timely for edge/IoT deployments and address practical deployment blockers (bias and staleness). Paper 1 is innovative for Fokker–Planck operator learning, but its impact is more specialized to computational physics/uncertainty quantification and may have narrower cross-field adoption.
Paper 2 is likely to have higher impact because it releases a large, harmonized, leakage-audited public clinical-genomic benchmark with locked tasks, standardized splits, and an evaluation harness—an enabling resource that can catalyze broad, reproducible work across oncology, ML, and precision medicine. It is timely (real-world evidence, longitudinal prediction, resistance in EGFR NSCLC) and has direct clinical relevance. Methodological rigor is strong in dataset curation and leakage control. Paper 1 is innovative but more niche (computational PDE/operator learning) and may have narrower immediate real-world uptake.
Paper 2 is likely higher impact due to broader applicability: efficient operator learning for Fokker–Planck equations can affect computational physics, uncertainty quantification, stochastic dynamics, and engineering. Its conditional normalizing-flow PINN with Chapman–Kolmogorov reformulation and handling of Dirac-delta initial conditions addresses a core numerical difficulty and could generalize to many PDE/SDE settings. Paper 1 is a solid applied ML contribution in EEG emotion recognition, but transformer-based EEG classifiers are a crowded area and the impact is more domain-specific. Methodological innovation and cross-field breadth favor Paper 2.
Paper 1 likely has higher impact because it introduces a general-purpose benchmark/test suite (ERBench) addressing a widely recognized evaluation gap in symbolic regression/equation discovery (robustness to noise, sampling regimes, dimensionality). Benchmarks often catalyze community-wide progress, standardize comparisons, and influence many downstream methods across ML and scientific computing. Paper 2 proposes a solid, timely method for Fokker–Planck operator learning, but it is narrower in scope (specific PDE class) and closer to incremental advances in PINNs/normalizing flows. Overall, Paper 1’s breadth and community-enabling nature suggest greater potential impact.
LargeMonitor introduces a novel paradigm for online task-free continual learning by decoupling drift detection and diagnosis using large pretrained models (LVMs and LMMs), addressing a fundamental gap in existing approaches. Its breadth of impact is significant—bridging foundation models with continual learning is timely and relevant to the rapidly growing AI community. Paper 2 presents a solid but more incremental contribution combining normalizing flows with PINNs for Fokker-Planck equations, addressing a narrower audience. Paper 1's novelty in leveraging LMMs for semantic diagnosis of distribution shifts and its broader applicability give it higher potential impact.
Paper 2 has higher estimated impact due to stronger timeliness and broader applicability: it connects foundation-model in-context learning with Bayesian optimization for molecule/protein design, a highly active area with clear real-world relevance. The proposed LSBO-specific continued pretraining and anchoring regularizer is a pragmatic, general recipe that could transfer to many latent-design settings and spur follow-on work across ML, optimization, and computational chemistry. Paper 1 is technically solid and novel for Fokker–Planck operator learning, but its impact is more specialized to stochastic PDE/SDE communities and narrower in immediate application scope.
Paper 2 likely has higher scientific impact due to strong timeliness and broad applicability in LLM post-training. Enabling on-policy distillation across different tokenizers removes a major practical constraint, expanding teacher–student pairing across model families and improving compute efficiency—highly relevant to current industry and academic workflows. The contribution is broadly impactful across NLP, systems, and model compression. Paper 1 is technically novel for stochastic PDE operator learning, but its impact is narrower (specialized to Fokker–Planck/SDE settings) and likely targets a smaller community, despite solid methodological rigor.
Paper 1 introduces a novel and technically sophisticated framework combining conditional normalizing flows with PINNs for solving Fokker-Planck equations across varying initial conditions. It addresses fundamental challenges in computational stochastic dynamics with broad applications in physics, engineering, and finance. The methodological innovations (Chapman-Kolmogorov reformulation, linearized SDE base distribution, time-weighted loss) are substantive and generalizable. Paper 2, while practically useful, represents an incremental contribution to performance estimation under distribution shift—fusing foundation and base model predictions—with narrower methodological novelty and more limited cross-disciplinary impact.
Paper 1 addresses a critical stability issue in GRPO, a highly relevant RL algorithm currently driving state-of-the-art LLM reasoning and instruction following. Its innovations in reward stabilization have immediate, widespread applications across AI development. Paper 2 presents a solid methodological advance in physics-informed neural networks for stochastic dynamics, but its impact is relatively confined to computational physics and applied mathematics, whereas Paper 1 affects the rapidly moving and broadly impactful field of large language models.