Generative sampling with physics-informed kernels

Friederike Ihssen; Jan M. Pawlowski; Renzo Kapust

REVIEW

Generative sampling with physics-informed kernels

Friederike Ihssen, Renzo Kapust, Jan M. Pawlowski

High Energy Physics - Lattice · Submitted 30 Oct 2026 · Published version · 27 references

Pith's verdict: Correct algebra and a useful residual-correction recipe, but the OOD-resolution claim is asserted rather than tested, and a zero-dimensional demo cannot test it.

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arXiv ID2610.41864 (version 1)

Pith IDHGA2SK3U, Pith's permanent record of this paper and its review

CitationFriederike Ihssen, Jan M. Pawlowski, Renzo Kapust. "Generative sampling with physics-informed kernels." arXiv:2610.41864 (2026). https://pith.science/paper/HGA2SK3U

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Editor's note

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Bottom line: the math is right and the residual-correction workflow is worth knowing about, but the headline claim — that PIKs structurally solve the two out-of-domain problems — is a statement about the architecture, not something this paper tests. The demonstration is zero-dimensional, and 0D cannot exhibit the curse of dimensionality the architecture is offered against.

What's real. Fixing S_t analytically does turn (7) into a linear first-order PDE for the kernel, and the layers really do decouple (b29, b116). That is by construction rather than discovery, and the paper more or less says as much. The nicer trick is (10)–(11): rewriting the normalisation flow as a PDE condition and taking differences of Wegner equations means you never have to sample or learn log N_t (b62). That is clean, and I had not seen it packaged this way. The RBF residual correction in Section 3.2 is the right kind of idea for this problem class. Figures 2 and 4 compare against HMC, which is an honest external check, though both are qualitative — no error bars, no acceptance rates.

Where I would push. First, the volume question, which the stress-test flags and which I think is the whole ballgame. Decoupling removes the coupling in t. It does not remove the need to represent and solve, at each t, a D-dimensional PDE for a D-component field. The size of the basis (12), the collocation choice, and the conditioning of A_t k_t = b_t are exactly where the cost lives at volume, and at one degree of freedom there is nothing to see. Section 3.2 concedes this as ongoing work, so it is an acknowledged gap rather than a hidden one — but it leaves the abstract's "structurally tackle out-of-domain problems" unverified. Second, Figure 3 shows the corrected kernel is closer to the truth, but never shows the resulting distribution improving, and the truth appears to be a solution of the same equation. Fine as a numerical demo; it says little about sampling.

Smaller notes. The architecture citation [8] is a PoS LATTICE2025 contribution with the same title and the same three authors. Provenance is stated honestly ("as constructed in [8]"), but a reader cannot check the construction independently from a conference companion. I would not hold [27] against the results; it is an outlook pointer to work in preparation, which is normal. The aside that other continuous models "must" solve (7) exactly at the last step to be "truly generative" is asserted rather than argued, and is the one place the prose outruns the content.

Who it is for: people already working on flow or diffusion samplers for lattice field theory who want the PIRG kernel picture in short form and the correction recipe. Not a standalone result.

Peer review: yes, send it out. As a proceedings it is a fair record of a correct construction with an honest limitations section, and the correction workflow deserves a referee's eye. Calibrate expectations though — the interesting test is the D>1 paper, not this one. I would cite [8] or [9] rather than this.

Referee report

Significance

If the stated claims hold, PIKs would be a genuinely useful architectural idea: per-layer solves are independent and linear, the normalisation constant is handled without a learned loss, and the residual equation gives a systematic, retraining-free route to refine a sampler. The algebraic core is transparent and the HMC comparisons in Figs. 2 and 4 provide an external check on the 0D implementation. I also note that the authors explicitly acknowledge in Section 3.2 that explicit volume scaling is ongoing research and that the correction example is internal to Eq. (11); these caveats are to their credit. The contribution is therefore best read as a proof-of-principle of the decoupling construction. The gap between that and the abstract's claim of structurally tackling out-of-domain problems is the main substantive issue: at one degree of freedom the construction is exact and the linear systems are trivial, so the volume-dependent behaviour that motivates the paper is not tested.

Major objections (3)

[Abstract and §3.2, Figs. 3–4] The headline claim that PIKs 'structurally tackle out-of-domain problems' is not supported by the numerical evidence presented. Section 3.2 explicitly defers 'the explicit volume scaling' to ongoing research, and the entire demonstration is at one degree of freedom. The residual-correction experiment in Fig. 3 verifies that an approximate kernel can be improved toward a solution of Eq. (11); it does not establish that solutions of Eq. (11) are representable in the basis (12) at L^d degrees of freedom, that the collocation matrices A_t are well-conditioned at that size, or that the flow ODE (4) remains practical. Because the OOD-resolution claim is the paper's motivation, the manuscript needs either a lattice demonstration (even at modest D greater than 1) with data on basis size, conditioning, and error, or a careful narrowing of the claim to a proof-of-principle.
[§2, Eqs. (7)–(12)] For D greater than 1, Eq. (7) is a single scalar linear PDE for the D components of v_t, so the reduction to 'an independent set of linear systems A_t k_t = b_t' (text after Eq. (12)) requires additional structure that is not specified in the paper: a vector-valued kernel basis, a well-posedness or gauge-fixing condition, and a choice of collocation pairs that makes the finite system determined and nonsingular. The zero-dimensional example has a scalar kernel and therefore does not exercise this step. Since this step is precisely where the curse of dimensionality enters, the layerwise decoupling in Eq. (6) is necessary but not sufficient for tractability at volume; the paper should state and justify the finite-dimensional reduction.
[§3.1–3.2] The numerical demonstrations omit the parameters needed to reproduce them: the number of basis functions M in Eq. (12), the frequencies omega_{j,t} and how they are chosen, the collocation points used to impose Eq. (11), the RBF support radii in Section 3.2, and the linear solver. No quantitative distributional error is reported for Fig. 2 or Fig. 3 (for example, acceptance rate, KL divergence, or moment errors). If these details appear in the companion paper [8], the proceedings should cite the specific equations; as it stands, the 'practical feasibility' of the independent-layer solves cannot be assessed or reproduced from this manuscript alone.

Minor objections (4)

[§2, Eq. (11)] The notation '... at phi = chi' is ambiguous because the second square bracket still displays partial_phi S_t(phi); please write the second term with chi-labelled derivatives, for example [∂/∂chi_i - ∂S_t(chi)/∂chi_i] v_t(chi), so the reader can see that the expression is evaluated at chi.
[§3.2] There is a typo in the opening sentence of Section 3.2: 'we know turn ot' should read 'we now turn to'.
[References] Reference [8] carries the same title and author list as this manuscript; please clarify in the text whether it is the companion full-length paper, so that the self-citation and the split of novel content are unambiguous.
[Fig. 2] With 10^6 samples, statistical uncertainties on the histogram bins are small but not negligible; showing them, or reporting a quantitative distance between the PIK and HMC distributions, would make the claimed 'excellent agreement' verifiable.

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The paper

The paper claims

This paper claims that fixing the action path $S_t$ analytically turns generative sampling into independent, analytically known linear solves, one per layer, with built-in residual correction — a structural answer to the two…

The reading

This paper claims that generative Monte-Carlo sampling can be rebuilt so that no transformation layer has to be learned from data: once the action path $S_t$ is chosen by hand, each layer's kernel — the vector field that pushes configurations forward — satisfies a known linear first-order PDE, the Wegner equation, and the kernel at one time step is independent of all other steps. The generative task then reduces to an independent set of linear systems $A_t k_t = b_t$, one per layer, which can be solved in parallel, refined by residual correction, and checked against the exact equation at any time. The authors argue that this structure answers the two out-of-domain failure modes of learned samplers, the curse of dimensionality and the curse of limited information, because the map is the solution of an analytically known equation rather than an interpolation trained on finite samples. In a zero-dimensional $\phi^4$ theory the modelled distribution matches hybrid Monte Carlo sampling, a deliberately imperfect kernel is corrected close to the exact one, and the magnetisation is obtained continuously along the flow. If the scheme survives at finite volume, the expensive final accept–reject correction of flow-based samplers would shrink to a controlled residual solve.

What carries the argument

The carrying object is the physics-informed RG pair $(S_t, \dot{\phi}_t)$: a chosen action path together with the flow kernel that transports configurations. The mechanism is the Wegner flow equation (3), rewritten for the action as (7), which for a fixed action path is a linear first-order PDE in the kernel; the decoupling property (6) makes each time step solvable on its own; the two-point difference (11) defines the linear operator $\mathcal{L}_\chi$ and sidesteps the normalisation integral; the kernel expansion (12) converts each layer into the linear system $A_t k_t = b_t$; and the residual equation (14), solved by collocation with compactly supported Wendland radial basis functions, supplies the systematic correction. The demonstration uses the straight-line action path (13) in $(m^2, \lambda)$ between a Gaussian base and a multimodal $\phi^4$ target, with a $\mathbb{Z}_2$-symmetric kernel basis $K_{1,t} = \phi$ and $K_{j,t} = \sin(\omega_{j,t}\phi)$ for $j \geq 2$.

Extended reading notes

Core claim

The central claim is the decoupling property of Eq. (6): if the generating function $F_t$ — here the action $S_t$ — is fully specified, the kernel $\dot{\phi}_t(\phi) = \dot{\phi}_t(\phi; F_t)$ is determined by that time step alone. With the action path fixed, the Wegner equation (7), $dS_t/dt = (\partial_{\phi_i} - \partial S_t/\partial \phi_i)\,\dot{\phi}_{t,i}$, is a linear first-order PDE for the kernel; expanding the kernel in a basis $\{K_{j,t}\}$ and imposing the two-point condition (11) turns every layer into an independent linear system $A_t k_t = b_t$ with analytically known coefficients. Two structural consequences carry the argument: the normalisation flow $d\log N_t/dt$ follows directly from (10), so the partition-function integral is never sampled, and linearity of the equation in the kernel allows any approximate solution to be corrected by a residual solve (14), iterated if needed. The demonstration at one degree of freedom shows agreement between the modelled and hybrid-Monte-Carlo distributions, a residual-corrected kernel approaching the exact one, and observables computed continuously from a single set of pushed samples.

Load-bearing premise

The load-bearing premise is that a hand-chosen action path with a truncated, time-independent operator basis is accurate enough that fixing $S_t$ genuinely decouples the layers: the decoupling is exact in the zero-dimensional demonstration, the operator time-dependence is dropped for this work (Section 2), and the paper itself states that explicit volume scaling, where the approximation would be tested, is part of ongoing research (Section 3.2).

Editorial extensions

If this is right

Layerwise kernels are solved once from analytically known linear systems, so the whole flow parallelises across the $N_t$ time steps and needs no global loss or backpropagation.
Because the Wegner equation is linear in the kernel, an imperfect or out-of-domain kernel can be improved locally by residual solves, giving systematic error control that the authors contrast with normalising flows and diffusion models.
The normalisation flow is computed from (10) without sampling the partition function, avoiding what the authors point out other learned interpolants must estimate or learn.
Observables can be evaluated continuously along the flow from a single batch of pushed samples, and parameter-conditional flows are obtained without inflating the network.
The same construction applies to other generating functions such as the rate function, and the authors identify complex actions and sign problems as the intended next targets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

At finite volume the truncated, time-independent operator basis is the likely bottleneck: on a lattice, $\partial \hat S_t / \partial \phi_i$ couples neighbouring sites, so a testable extension is to let the basis flow with $t$ or to choose the action path so that the kernel representation closes — that truncation error is exactly where the decoupling (6) becomes approximate.
The residual loop (14) is structurally classical iterative refinement for PDE collocation; comparing its convergence with retraining the kernel on fresh samples would quantify how much the analytic equation contributes beyond the data.
If the decoupling holds at scale, wall-clock sampling cost shifts from data-limited interpolation to parallel linear algebra, and a PIK sampler would function more like a physics-informed preconditioner for MCMC than like a trained network.

Original abstract

We construct a generative network for Monte-Carlo sampling in lattice field theories and beyond, for which the layerwise propagation is computed and optimised independently on each layer. The architecture uses physics-informed renormalisation group flows that provide access to the layerwise propagation step from one layer to the next in terms of a simple first order partial differential equation for the respective renormalisation group kernel through a given layer. Thus, it transforms the generative task into that of solving once the set of independent and linear differential equations for the kernels of the transformation. As these equations are analytically known, the kernels can be refined iteratively. This allows us to structurally tackle out-of-domain problems generally encountered in generative models and opens the path to further optimisation. We illustrate the practical feasibility of the architecture within simulations in scalar field theories.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-02: 1 request, 4,917 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.26 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.05 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We construct a generative network for Monte-Carlo sampling in lattice field theories and beyond, for which the layerwise propagation is computed and optimised independently on each layer. The architecture uses physics-informed renormalisation group flows that provide access to the layerwise propagation step from one layer to the next in terms of a simple first order partial differential equation for the respective renormalisation group kernel through a given layer. Thus, it transforms the generative task into that of solving once the set of independent and linear differential equations for the kernels of the transformation. As these equations are analytically known, the kernels can be refined iteratively. This allows us to structurally tackle out-of-domain problems generally encountered in generative models and opens the path to further optimisation. We illustrate the practical feasibility of the architecture within simulations in scalar field theories.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We construct a generative network for Monte-Carlo sampling in lattice field theories and beyond, for which the layerwise propagation is computed and optimised independently on each layer.	contribution 0.94	0.93	0.19	asserted 1.00
s1	The architecture uses physics-informed renormalisation group flows that provide access to the layerwise propagation step from one layer to the next in terms of a simple first order partial differential equation for the respective renormalisation group kernel through a given layer.	method 0.98	0.73	0.09	asserted 1.00
s2	Thus, it transforms the generative task into that of solving once the set of independent and linear differential equations for the kernels of the transformation.	method 0.85	0.71	0.19	asserted 1.00
s3	As these equations are analytically known, the kernels can be refined iteratively.	method 0.93	0.54	0.09	asserted 0.99
s4	This allows us to structurally tackle out-of-domain problems generally encountered in generative models and opens the path to further optimisation.	interpretation 0.98	0.55	0.40	asserted 0.84
s5	We illustrate the practical feasibility of the architecture within simulations in scalar field theories.	result 0.47	0.93	0.12	asserted 0.99

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.92.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Setup of the PIK-architecture, with fully accessible the path of 𝐹𝑡 (black) and kernels 𝜙¤ 𝑡 (green) with completely independent parameters 𝑘𝑡 (red). −4 −2 0 2 4 φ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 pt=1(φ) PIK HMC

**Figure 2.** shows a comparison of the targeted (orange) and modelled (black) distribution at the final time 𝑡 = 1. Here, the targeted distribution is obtained using 106 samples from a standard hybrid Monte Carlo algorithm (HMC). The modelled distribution was generating by pushing 106 samples from the initial Gaußian distribution through the PIK flow. We find an excellent agreement between both distributions. 3.2 OOD resolution Having showcased that PIKs have the capacity to compute each kernel 𝜙¤ 𝑡 independently without the requirement of computing the change in the normalisation, we know turn o

**Figure 3.** Comparison of the true (orange), the imperfect (black), and the corrected (green) kernel 𝜙¤ 𝑡=1 for the zero-dimensional 𝜙 4 -theory. 0.0 0.2 0.4 0.6 0.8 1.0 t 0.8 1.0 1.2 1.4 1.6 h| M(t)|i PIK HMC

Showing 3 of the paper's 4 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.41864 by the authors.

References

Works cited (27)

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R. Kapust, F. Ihssen and J.M. Pawlowski,Generative sampling with physics-informed kernels,PoSLATTICE2025(2026) 055 [2510.26678]

View on Pith arXiv 2026
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F. Ihssen, R. Kapust and J.M. Pawlowski,Solving real sign problems,in preparation(2026) . 10

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F. Ihssen and J.M. Pawlowski,Physics-informed renormalisation group flows,Annals Phys. 481(2025) 170177 [2409.13679]

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B. Alles, G. Boyd, M. D’Elia, A. Di Giacomo and E. Vicari,Hybrid Monte Carlo and topological modes of full QCD,Phys. Lett. B389(1996) 107 [hep-lat/9607049]

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L. Del Debbio, G.M. Manca and E. Vicari,Critical slowing down of topological modes, Phys. Lett. B594(2004) 315 [hep-lat/0403001]

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M. Luscher,Trivializing maps, the Wilson flow and the HMC algorithm,Commun. Math. Phys.293(2010) 899 [0907.5491]

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D.J. Rezende and S. Mohamed,Variational Inference with Normalizing Flows, 5, 2015 [1505.05770]

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M.S. Albergo, G. Kanwar and P.E. Shanahan,Flow-based generative models for Markov chain Monte Carlo in lattice field theory,Phys. Rev. D100(2019) 034515 [1904.12072]

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K.A. Nicoli, C.J. Anders, L. Funcke, T. Hartung, K. Jansen, P. Kessel et al.,Estimation of Thermodynamic Observables in Lattice Field Theories with Deep Generative Models,Phys. Rev. Lett.126(2021) 032001 [2007.07115]

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K.A. Nicoli, C.J. Anders, T. Hartung, K. Jansen, P. Kessel and S. Nakajima,Detecting and mitigating mode-collapse for flow-based sampling of lattice field theories,Phys. Rev. D108 (2023) 114501 [2302.14082]

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [8]

R. Kapust, F. Ihssen and J.M. Pawlowski,Generative sampling with physics-informed kernels,PoSLATTICE2025(2026) 055 [2510.26678]

View on Pith arXiv 2026

[2] [27]

F. Ihssen, R. Kapust and J.M. Pawlowski,Solving real sign problems,in preparation(2026) . 10

View on Pith 2026

[3] [9]

F. Ihssen and J.M. Pawlowski,Physics-informed renormalisation group flows,Annals Phys. 481(2025) 170177 [2409.13679]

View on Pith Pith review arXiv 2025

[4] [1]

U. Wolff,CRITICAL SLOWING DOWN,Nucl. Phys. B Proc. Suppl.17(1990) 93

View on Pith 1990

[5] [2]

B. Alles, G. Boyd, M. D’Elia, A. Di Giacomo and E. Vicari,Hybrid Monte Carlo and topological modes of full QCD,Phys. Lett. B389(1996) 107 [hep-lat/9607049]

View on Pith Pith review arXiv 1996

[6] [3]

L. Del Debbio, G.M. Manca and E. Vicari,Critical slowing down of topological modes, Phys. Lett. B594(2004) 315 [hep-lat/0403001]

View on Pith Pith review arXiv 2004

[7] [4]

M. Luscher,Trivializing maps, the Wilson flow and the HMC algorithm,Commun. Math. Phys.293(2010) 899 [0907.5491]

View on Pith Pith review arXiv 2010

[8] [5]

D.J. Rezende and S. Mohamed,Variational Inference with Normalizing Flows, 5, 2015 [1505.05770]

View on Pith Pith review arXiv 2015

[9] [6]

M.S. Albergo, G. Kanwar and P.E. Shanahan,Flow-based generative models for Markov chain Monte Carlo in lattice field theory,Phys. Rev. D100(2019) 034515 [1904.12072]

View on Pith Pith review arXiv 2019

[10] [7]

K.A. Nicoli, C.J. Anders, L. Funcke, T. Hartung, K. Jansen, P. Kessel et al.,Estimation of Thermodynamic Observables in Lattice Field Theories with Deep Generative Models,Phys. Rev. Lett.126(2021) 032001 [2007.07115]

View on Pith Pith review arXiv 2021

[11] [10]

F.J. Wegner,Some invariance properties of the renormalization group,J. Phys. C7(1974) 2098

View on Pith 1974

[12] [11]

J. Cotler and S. Rezchikov,Renormalizing Diffusion Models,2308.12355. 9 Generative sampling with physics-informed kernelsRenzo Kapust

View on Pith Pith review arXiv

[13] [12]

M.S. Albergo and E. Vanden-Eijnden,NETS: A Non-Equilibrium Transport Sampler, 2410.02711

View on Pith Pith review arXiv

[14] [13]

B. Máté and F. Fleuret,Learning Interpolations between Boltzmann Densities,2301.07388

View on Pith Pith review arXiv

[15] [14]

D.C. Hackett, C.-C. Hsieh, S. Pontula, M.S. Albergo, D. Boyda, J.-W. Chen et al., Flow-based sampling for multimodal and extended-mode distributions in lattice field theory, 2107.00734

View on Pith Pith review arXiv

[16] [15]

S. Chen, O. Savchuk, S. Zheng, B. Chen, H. Stoecker, L. Wang et al.,Fourier-flow model generating Feynman paths,Phys. Rev. D107(2023) 056001 [2211.03470]

View on Pith Pith review arXiv 2023

[17] [16]

K.A. Nicoli, C.J. Anders, T. Hartung, K. Jansen, P. Kessel and S. Nakajima,Detecting and mitigating mode-collapse for flow-based sampling of lattice field theories,Phys. Rev. D108 (2023) 114501 [2302.14082]

View on Pith Pith review arXiv 2023

[18] [17]

J. Köhler, L. Klein and F. Noé,Equivariant flows: Exact likelihood generative learning for symmetric densities, 2020

View on Pith 2020

[19] [18]

M. Gerdes, P. de Haan, C. Rainone, R. Bondesan and M.C.N. Cheng,Learning lattice quantum field theories with equivariant continuous flows,SciPost Phys.15(2023) 238 [2207.00283]

View on Pith Pith review arXiv 2023

[20] [19]

R.T.Q. Chen, Y. Rubanova, J. Bettencourt and D. Duvenaud,Neural Ordinary Differential Equations,1806.07366

View on Pith Pith review arXiv

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