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arxiv: 2306.10407 · v3 · submitted 2023-06-17 · 💻 cs.LG · cs.AI· physics.bio-ph· q-bio.CB

FP-IRL: Fokker--Planck Inverse Reinforcement Learning -- A Physics-Constrained Approach to Markov Decision Processes

Chengyang Huang , Siddhartha Srivastava , Kenneth K. Y. Ho , Kathy E. Luker , Gary D. Luker , Xun Huan , Krishna Garikipati This is my paper

Pith reviewed 2026-05-24 08:24 UTC · model grok-4.3

classification 💻 cs.LG cs.AIphysics.bio-phq-bio.CB
keywords inverse reinforcement learningFokker-Planck dynamicsMarkov decision processvariational system identificationphysics-constrained learningtrajectory inferencereward function recovery
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The pith

FP-IRL infers both reward and transition functions directly from trajectory data by mapping MDPs to Fokker-Planck dynamics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Most inverse reinforcement learning methods need the transition function known or estimated in advance, which limits their use when dynamics are unknown. FP-IRL removes this requirement by establishing a correspondence that equates reward maximization in an MDP with free energy minimization under Fokker-Planck dynamics. From observed trajectories the method first infers the underlying potential function through variational system identification. Analytic expressions then recover the full set of MDP elements: reward, transition probabilities, and policy. The approach therefore works on systems where direct sampling of transitions is impossible while keeping the inferred quantities physically interpretable.

Core claim

A correspondence between Markov decision processes and the Fokker-Planck equation links reward maximization in the MDP to free energy minimization in the FP dynamics. Inference of the FP potential function from trajectory data via variational system identification then yields analytic expressions for the reward function, transition function, and policy, allowing all three to be recovered simultaneously without prior access to sampled transitions.

What carries the argument

The MDP-FP correspondence that equates reward maximization with free energy minimization, enabling analytic recovery of reward, transition, and policy from a single inferred potential function.

If this is right

  • Inverse reinforcement learning becomes feasible in domains where transition dynamics cannot be sampled or estimated beforehand.
  • Reward, transition, and policy are recovered together and remain consistent with one another through the shared potential.
  • The inferred quantities retain physical meaning because they derive from an FP potential.
  • The method applies to both synthetic test cases and continuous control tasks such as a modified Mountain Car environment.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same physics-constrained inference route could be tested on data from diffusion processes that only approximately obey Fokker-Planck equations.
  • If the analytic recovery step holds, similar mappings might be sought between other reinforcement learning objectives and known physical evolution equations.
  • The framework naturally supplies a consistency check: the inferred transition function can be validated against any available partial observations of state changes.

Load-bearing premise

The system dynamics must follow Fokker-Planck equations and the MDP-FP correspondence must permit exact analytic recovery of all components from the potential.

What would settle it

The recovered reward and transition functions, when inserted back into the MDP, produce trajectories that differ systematically from the original observed data.

Figures

Figures reproduced from arXiv: 2306.10407 by Chengyang Huang, Gary D. Luker, Kathy E. Luker, Kenneth K. Y. Ho, Krishna Garikipati, Siddhartha Srivastava, Xun Huan.

Figure 1
Figure 1. Figure 1: Comparison of inferred ground truth value and reward (using highest resolution mesh with [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) KL divergence DKL(pt|qt) of the probability distribution between data distribution and simulation probability distribution over time. The errors of the (b) value function and (c) partial derivatives of the value function, estimated as 1 |Ω| R Ω (f(x) − fGT(x))2dx 1/2 truth state-action value generated using fixed-point iteration (details provided in Appendix B). The results of the convergence analysis… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Kymograph for the cancer cell migration data: each column shows the different vari [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
read the original abstract

Inverse reinforcement learning (IRL) is a powerful paradigm for uncovering the incentive structure that drives agent behavior, by inferring an unknown reward function from observed trajectories within a Markov decision process (MDP). However, most existing IRL methods require access to the transition function, either prescribed or estimated \textit{a priori}, which poses significant challenges when the underlying dynamics are unknown, unobservable, or not easily sampled. We propose Fokker--Planck inverse reinforcement learning (FP-IRL), a novel physics-constrained IRL framework tailored for systems that can be described by Fokker--Planck (FP) dynamics. FP-IRL simultaneously infers both the reward and transition functions directly from trajectory data, without requiring access to sampled transitions. Our method leverages a correspondence between MDPs and the FP equation, linking reward maximization in MDPs with free energy minimization in FP dynamics. This connection enables inference of the FP potential function using our inference approach of variational system identification, from which the full set of MDP components -- reward, transition, and policy -- can be recovered using analytic expressions. We demonstrate the effectiveness of FP-IRL through experiments on synthetic benchmarks and a modified version of the Mountain Car problem. Our results show that FP-IRL achieves accurate recovery of agent incentives while preserving computational efficiency and physical interpretability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes FP-IRL, a physics-constrained IRL method for MDPs whose dynamics admit a Fokker-Planck description. It asserts a direct correspondence between MDP reward maximization and FP free-energy minimization, uses variational system identification to recover the FP potential from trajectories alone, and then applies analytic expressions to extract the reward function, transition kernel, and policy without requiring sampled transitions. Experiments on synthetic benchmarks and a modified Mountain Car task are used to illustrate recovery accuracy and efficiency.

Significance. If the asserted MDP-FP correspondence is exact, parameter-free, and yields unique analytic recoveries, the framework would enable IRL in continuous-state systems with unknown dynamics while preserving physical interpretability; this would be a distinctive contribution relative to standard IRL methods that presuppose or estimate the transition function.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (MDP-FP correspondence): the central claim that reward maximization in an MDP is exactly equivalent to free-energy minimization under FP dynamics, permitting closed-form recovery of r(s,a), P(s'|s,a), and π from the inferred potential alone, is load-bearing. The manuscript must supply the explicit derivation (including the role of the diffusion coefficient and action embedding) showing that the mapping is bijective and independent of quantities fitted inside the variational procedure; without it the simultaneous inference guarantee does not follow.
  2. [§4] §4 (variational system identification and analytic recovery formulas): the expressions that convert the recovered FP potential into the MDP reward and transition must be shown to be free of hidden parameters or discretization artifacts. If the recovery formulas implicitly re-introduce knowledge of the FP operator or require additional constraints on the action space, the claim that both reward and transition are inferred directly from trajectories is undermined.
minor comments (2)
  1. [Experiments] The modified Mountain Car experiment should explicitly state how the continuous FP dynamics are reconciled with the discrete action set and whether any approximation artifacts affect the reported recovery accuracy.
  2. [Notation] Notation for the FP potential and the MDP components should be unified across sections to avoid ambiguity when mapping between the two formalisms.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The two major comments both concern the need for fuller explicit derivations and parameter-independence arguments in Sections 3 and 4. We address each point below and will incorporate the requested material in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (MDP-FP correspondence): the central claim that reward maximization in an MDP is exactly equivalent to free-energy minimization under FP dynamics, permitting closed-form recovery of r(s,a), P(s'|s,a), and π from the inferred potential alone, is load-bearing. The manuscript must supply the explicit derivation (including the role of the diffusion coefficient and action embedding) showing that the mapping is bijective and independent of quantities fitted inside the variational procedure; without it the simultaneous inference guarantee does not follow.

    Authors: We agree that an expanded, self-contained derivation is necessary to make the bijectivity and independence claims fully transparent. Section 3 already sketches the MDP-to-FP correspondence, but the revised manuscript will enlarge this section with a complete step-by-step derivation that explicitly treats the diffusion coefficient and the action-embedding map. A short appendix will supply the bijectivity argument and verify that the recovered quantities do not depend on any parameters internal to the variational procedure. revision: yes

  2. Referee: [§4] §4 (variational system identification and analytic recovery formulas): the expressions that convert the recovered FP potential into the MDP reward and transition must be shown to be free of hidden parameters or discretization artifacts. If the recovery formulas implicitly re-introduce knowledge of the FP operator or require additional constraints on the action space, the claim that both reward and transition are inferred directly from trajectories is undermined.

    Authors: We concur that the analytic recovery formulas require an explicit demonstration of parameter independence. In the revision we will augment Section 4 with a dedicated subsection that (i) states all assumptions on the action space, (ii) shows that the conversion expressions contain no hidden parameters or re-introduced FP-operator terms, and (iii) confirms the absence of discretization artifacts under the continuous-state formulation used in the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain.

full rationale

The paper's derivation rests on leveraging an asserted correspondence between MDPs and FP dynamics to connect reward maximization with free-energy minimization, followed by variational inference of the potential and analytic recovery of reward/transition/policy. No quoted equations or steps in the abstract or description reduce any claimed prediction to fitted inputs by construction, nor does any load-bearing premise collapse to a self-citation chain or ansatz smuggled from prior author work. The variational system identification and analytic expressions are presented as independent of the target quantities, and the method is tested on external benchmarks, making the chain self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on the domain assumption that the system obeys Fokker-Planck dynamics and that an exact analytic correspondence to MDP components exists. No free parameters or invented entities are identifiable from the abstract alone.

axioms (1)
  • domain assumption The system dynamics admit a Fokker-Planck description and a direct correspondence exists between MDP reward maximization and FP free-energy minimization.
    Explicitly stated as the tailoring condition and enabling link in the abstract.

pith-pipeline@v0.9.0 · 5808 in / 1393 out tokens · 23355 ms · 2026-05-24T08:24:36.022113+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Conjecture 3.1. The potential function in FP is equivalent to the negative state-action value function in MDP: ψ(s,a)=−Q^π(s,a).

  • IndisputableMonolith/Foundation/BranchSelection.lean branch_selection echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    F(p) = ∫ ψ(x)p(x)dx + β⁻¹ ∫ p(x)log p(x)dx … the solution of pt+1 = arg min W₂(pt,p)² + Δt F(p) converges to the FP PDE

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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