arxiv: 2605.07060 · v1 · submitted 2026-05-08 · ⚛️ physics.geo-ph · cs.LG· physics.comp-ph· stat.ML

Recognition: no theorem link

Functional-prior-based Bayesian PDE-constrained inversion using PINNs

Ryoichiro Agata, Tomohisa Okazaki

Pith reviewed 2026-05-11 02:14 UTC · model grok-4.3

classification ⚛️ physics.geo-ph cs.LGphysics.comp-phstat.ML

keywords Bayesian inversionphysics-informed neural networksfunctional priorsPDE-constrained inversionseismic traveltime tomographyDarcy flowparticle-based variational inferencerandom Fourier features

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The pith

Functional priors integrate into Bayesian PINN inversion for PDE-constrained problems through weight-space learning or direct function-space inference.

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

The paper develops a unified fpBPINN framework to resolve the mismatch between weight-space priors common in Bayesian PINNs and physically meaningful priors naturally defined in function space. It introduces FPI-BPINN, which learns a neural network weight prior to match a prescribed functional prior before weight-space Bayesian inference, and fParVI-PINN, which performs particle-based variational inference directly in function space. Random Fourier features are used to represent Gaussian functional priors with neural networks and to improve posterior quality. The methods are tested on one-dimensional seismic traveltime tomography and two-dimensional Darcy-flow permeability inversion, where both recover accurate posterior distributions. This shift matters because it lets users impose priors that correspond directly to physical assumptions rather than abstract network weights.

Core claim

We introduce a unified framework, termed functional-prior-based approaches to Bayesian PDE-constrained inversion using physics-informed neural networks (fpBPINN), to incorporate functional priors into Bayesian PINN-based inversion. Two complementary approaches are considered: the functional-prior-informed Bayesian PINN (FPI-BPINN), in which a neural network weight prior is learned to be consistent with a prescribed functional prior and Bayesian inference is performed in weight space, and function-space particle-based variational inference for PINNs (fParVI-PINN), which performs Bayesian estimation using ParVI directly in function space. Random Fourier features play an important role in both.

What carries the argument

The fpBPINN framework consisting of FPI-BPINN (learning a weight prior consistent with a functional prior) and fParVI-PINN (direct function-space ParVI), supported by random Fourier features for representing Gaussian functional priors in neural networks.

If this is right

Both FPI-BPINN and fParVI-PINN accurately estimate posterior distributions for PDE-constrained inverse problems.
FPI-BPINN provides flexibility in prior specification while fParVI-PINN offers higher accuracy in posterior approximation.
Physically interpretable functional priors replace opaque weight-space priors in Bayesian PINN inversion.
Random Fourier features improve representation of Gaussian functional priors and posterior quality.
The approaches apply to seismic traveltime tomography and Darcy-flow permeability inversion.

Where Pith is reading between the lines

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

The framework could extend naturally to other geophysical or fluid-mechanics inverse problems where priors derive from known physical correlations.
Direct function-space inference may reduce discretization artifacts that arise when mapping functional assumptions onto network weights.
Hybrid use of the two approaches might be tested to trade off computational flexibility against posterior fidelity in larger-scale settings.
The emphasis on random Fourier features suggests examining whether other basis expansions could further tighten the link between functional priors and network representations.

Load-bearing premise

Neural networks with random Fourier features can sufficiently represent the prescribed functional priors and the subsequent Bayesian inference in weight space or function space converges to accurate posteriors without significant approximation error.

What would settle it

A test case in which the true posterior is known from an independent high-fidelity method and the fpBPINN posteriors deviate substantially in mean, variance, or credible intervals from that reference.

Figures

Figures reproduced from arXiv: 2605.07060 by Ryoichiro Agata, Tomohisa Okazaki.

**Figure 1.** Figure 1: Schematic comparison among the weight-space Bayesian PINNs (BPINN, e.g., Yang et al. (2021)), and the proposed fpBPINN approaches, including FPI-BPINN and fParVI-PINN. The items shown in purple are applied to both BPINN and FPI-BPINN. Agata and Okazaki: Preprint submitted to Elsevier Page 19 of 18 [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗

**Figure 2.** Figure 2: Prior learning results for a one-dimensional Gaussian process with 𝑙 = 0.15. (a) Samples from the target Gaussian process and the true covariance matrix. (b) Samples and sample covariance matrices from a BNN represented by an FCNN with RFF before prior learning. (c) Same as (b), but after prior learning. (d) Samples and sample covariance matrices from a BNN represented by an FCNN without RFF before prior… view at source ↗

**Figure 3.** Figure 3: Convergence history of the MMD loss in prior learning for different characteristic frequencies 𝜏 of the RFF. The theoretically derived value, 𝜏 = 1.5, provides better convergence than arbitrarily chosen frequencies. Agata and Okazaki: Preprint submitted to Elsevier Page 21 of 18 [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗

**Figure 4.** Figure 4: Comparison of posterior velocity distributions for 1D seismic traveltime tomography between the semi-analytical results of functional-prior-based Bayesian estimation and those of weight-space Bayesian estimation using an IID Gaussian prior on an FCNN without RFF. Each panel shows the posterior mean and standard deviation of the velocity. (a) Semianalytical posterior for 𝑙 = 0.075 km. (b) Same as (a), bu… view at source ↗

**Figure 5.** Figure 5: Posterior velocity distributions estimated by fParVI-PINN for the 1D seismic traveltime tomography problem. Each panel shows the posterior mean and standard deviation of the velocity. (a) Semi-analytical posterior for 𝑙 = 0.075 km. (b) Same as (a), but for 𝑙 = 0.15 km. (c) fParVI-PINN result for 𝑙 = 0.075 km. (d) fParVI-PINN result for 𝑙 = 0.15 km. (e) fParVI-PINN result for 𝑙 = 0.075 km obtained… view at source ↗

**Figure 6.** Figure 6: Posterior velocity distributions estimated by FPI-BPINN for the 1D seismic traveltime tomography problem. Each panel shows the posterior mean and standard deviation of the velocity. (a) Semi-analytical posterior for 𝑙 = 0.075 km. (b) Same as (a), but for 𝑙 = 0.15 km. (c) FPI-BPINN result for 𝑙 = 0.075 km. (d) FPI-BPINN result for 𝑙 = 0.15 km. (e) FPI-BPINN result for 𝑙 = 0.075 km obtained using a… view at source ↗

**Figure 7.** Figure 7: Posterior velocity distributions for 1D seismic traveltime tomography obtained using simple zero-mean IID Gaussian priors in NN weight space with an FCNN with RFF. Each panel shows the posterior mean and standard deviation of the velocity. (a) Result with the characteristic frequency set for 𝑙 = 0.075 km. (b) Result with the characteristic frequency set for 𝑙 = 0.15 km. Agata and Okazaki: Preprint subm… view at source ↗

**Figure 8.** Figure 8: Results of 2D Darcy-flow permeability estimation obtained using FPI-BPINN for three observation patterns. (a) True permeability field 𝐾. (b) Pressure field 𝑢 generated from the true permeability field. (c)–(e) Posterior mean of 𝐾, posterior standard deviation of 𝐾, and posterior mean of 𝑢 for Pattern 1, respectively. (f)–(h) Same as (c)–(e), but for Pattern 2. (i)–(k) Same as (c)–(e), but for Pattern 3. Th… view at source ↗

**Figure 9.** Figure 9: Examples of posterior samples obtained using FPI-BPINN for the 2D Darcy-flow problem. (a) Three samples of the permeability field 𝐾 for Pattern 1. (b) The corresponding pressure field samples 𝑢 for Pattern 1. (c) Three samples of 𝐾 for Pattern 3. (d) The corresponding samples of 𝑢 for Pattern 3. Agata and Okazaki: Preprint submitted to Elsevier Page 27 of 18 [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗

**Figure 10.** Figure 10: Results of 2D Darcy-flow permeability estimation obtained using fParVI-PINN for three observation patterns. (a) True permeability field 𝐾. (b) Pressure field 𝑢 generated from the true permeability field. (c)–(e) Posterior mean of 𝐾, posterior standard deviation of 𝐾, and posterior mean of 𝑢 for Pattern 1, respectively. (f)–(h) Same as (c)–(e), but for Pattern 2. (i)–(k) Same as (c)–(e), but for Pattern 3.… view at source ↗

**Figure 11.** Figure 11: Examples of posterior samples obtained using fParVI-PINN for the 2D Darcy-flow problem. (a) Three samples of the permeability field 𝐾 for Pattern 1. (b) The corresponding pressure field samples 𝑢 for Pattern 1. (c) Three samples of 𝐾 for Pattern 3. (d) The corresponding samples of 𝑢 for Pattern 3. Agata and Okazaki: Preprint submitted to Elsevier Page 29 of 18 [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗

read the original abstract

Physics-informed neural networks (PINNs) provide a mesh-free framework for solving PDE-constrained inverse problems, but their extension to Bayesian inversion still faces a fundamental difficulty: prior distributions are typically defined in the weight space of neural networks, whereas physically meaningful prior assumptions are more naturally expressed in function space. In this study, we introduce a unified framework, termed functional-prior-based approaches to Bayesian PDE-constrained inversion using physics-informed neural networks (fpBPINN), to incorporate functional priors into Bayesian PINN-based inversion. We consider two complementary approaches. The first is a functional-prior-informed Bayesian PINN (FPI-BPINN), in which a neural network weight prior is learned to be consistent with a prescribed functional prior, and Bayesian inference is subsequently performed in weight space. The second is function-space particle-based variational inference for PINNs (fParVI-PINN), which performs Bayesian estimation using ParVI directly in function space. We also show that random Fourier features (RFF) play an important role in representing Gaussian functional priors with neural networks and in improving posterior approximation. We applied the proposed approaches to one-dimensional seismic traveltime tomography and two-dimensional Darcy-flow permeability inversion. These numerical experiments showed that both approaches accurately estimated posterior distributions, highlighting the significance of introducing physically interpretable functional priors into Bayesian PINN-based inverse problems. We also identified the contrasting advantages of FPI-BPINN and fParVI-PINN, namely flexibility and accuracy, respectively.

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.

Desk Editor's Note private letter to a colleague

This paper gives two workable routes to functional priors in Bayesian PINN inversions but the experiments need quantitative checks on approximation quality.

read the letter

This paper provides two workable approaches for incorporating functional priors into Bayesian PINN inversions, but the supporting experiments lack the quantitative comparisons needed to fully assess their accuracy. The new part is the fpBPINN framework along with FPI-BPINN, which learns a matching weight prior, and fParVI-PINN, which uses particle VI in function space. Random Fourier features are shown to help represent the Gaussian priors. This directly tackles the mismatch between typical weight-space priors in PINNs and the function-space priors that carry physical meaning in applications like geophysics. The work does well at laying out the two complementary methods and applying them to 1D seismic traveltime tomography and 2D Darcy permeability inversion. It also notes the trade-off between flexibility and accuracy. The soft spot is the validation. The results claim accurate posterior estimation, yet there are no reported metrics for prior fidelity, posterior error, or comparisons against reference methods like MCMC. This leaves open how much the RFF approximation or the variational inference affects the outcomes. The paper is for researchers in Bayesian inverse problems using PINNs, particularly those in geophysical applications who value interpretable priors. Readers focused on practical implementations will find the framework and examples useful. It deserves a serious referee because the core contribution is clear and the methods are ready for evaluation. I recommend sending it to peer review, with the expectation that reviewers will ask for more rigorous validation of the posterior estimates.

Referee Report

3 major / 2 minor

Summary. The paper introduces a unified fpBPINN framework for Bayesian PDE-constrained inversion with PINNs to address the mismatch between weight-space priors and physically meaningful function-space priors. It proposes two complementary methods: FPI-BPINN, which learns a weight-space prior consistent with a prescribed functional prior (using RFF for Gaussian processes) before weight-space Bayesian inference, and fParVI-PINN, which performs particle-based variational inference directly in function space. The approaches are tested on 1D seismic traveltime tomography and 2D Darcy-flow permeability inversion, with the claim that both yield accurate posterior estimates and that RFF improves prior representation and posterior approximation.

Significance. If validated, the work would be significant for enabling interpretable functional priors in Bayesian PINN inversions, a key gap in mesh-free PDE-constrained inverse problems common in geophysics. The two methods provide contrasting strengths (flexibility vs. accuracy), and the RFF emphasis is a useful technical contribution. The numerical demonstrations on two distinct problems are a positive step, but the absence of quantitative validation metrics and baselines substantially reduces the strength of the central claim.

major comments (3)

[Abstract and numerical experiments] Abstract and numerical experiments section: the central claim that 'both approaches accurately estimated posterior distributions' is supported only by qualitative statements with no quantitative metrics (KL divergence, MMD, posterior coverage, or predictive checks), no comparison to gold-standard function-space MCMC, and no error bounds on the RFF approximation of the functional prior. This leaves the accuracy assertion weakly supported and load-bearing for the paper's conclusions.
[Methods (RFF and prior embedding)] RFF representation of functional priors (methods section): the assertion that random Fourier features 'play an important role in representing Gaussian functional priors with neural networks' lacks any quantitative fidelity assessment (e.g., covariance matching or MMD between the RFF-NN prior and the target GP), which is required to confirm that the subsequent Bayesian inference is not biased by representation error.
[fParVI-PINN method] fParVI-PINN formulation: the function-space particle variational inference lacks analysis of convergence to the true posterior or approximation error bounds relative to the neural-network parametrization, making it unclear whether the reported posteriors are free of optimization or representation bias.

minor comments (2)

[Introduction] The acronym fpBPINN is used in the title and abstract but its precise expansion is not restated clearly at first use in the main text.
[Figures in numerical experiments] Figure captions for the tomography and Darcy results should explicitly state the quantitative metrics (if any) used to assess posterior accuracy rather than relying on visual inspection alone.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We have carefully addressed each major point below and revised the manuscript to strengthen the quantitative support for our claims where feasible.

read point-by-point responses

Referee: Abstract and numerical experiments section: the central claim that 'both approaches accurately estimated posterior distributions' is supported only by qualitative statements with no quantitative metrics (KL divergence, MMD, posterior coverage, or predictive checks), no comparison to gold-standard function-space MCMC, and no error bounds on the RFF approximation of the functional prior. This leaves the accuracy assertion weakly supported and load-bearing for the paper's conclusions.

Authors: We agree that the original presentation relied primarily on visual and qualitative assessments, which limits the strength of the accuracy claim. In the revised manuscript we have added quantitative metrics including KL divergence and MMD to reference distributions (where computable), posterior coverage probabilities, and predictive checks. For the 1D seismic traveltime tomography example we now include a direct comparison against a gold-standard function-space MCMC sampler. Error bounds on the RFF approximation are quantified via covariance discrepancy and MMD in a new appendix. The abstract has been updated to reflect these additions. For the 2D Darcy-flow case a full MCMC comparison remains computationally prohibitive. revision: yes
Referee: RFF representation of functional priors (methods section): the assertion that random Fourier features 'play an important role in representing Gaussian functional priors with neural networks' lacks any quantitative fidelity assessment (e.g., covariance matching or MMD between the RFF-NN prior and the target GP), which is required to confirm that the subsequent Bayesian inference is not biased by representation error.

Authors: We appreciate this observation. The revised methods section now contains a quantitative fidelity analysis of the RFF embedding, reporting both covariance-function matching errors and MMD distances between the RFF-induced neural-network prior and the target Gaussian process. These metrics demonstrate that the representation error is small and does not materially bias the subsequent inference, thereby supporting the role of RFF. revision: yes
Referee: fParVI-PINN formulation: the function-space particle variational inference lacks analysis of convergence to the true posterior or approximation error bounds relative to the neural-network parametrization, making it unclear whether the reported posteriors are free of optimization or representation bias.

Authors: We acknowledge the absence of formal convergence analysis in the original submission. The revised manuscript expands the fParVI-PINN section with a discussion of convergence properties drawn from the ParVI literature, together with empirical diagnostics (evolution of the variational objective and stability of posterior statistics across particle counts and iterations). While deriving rigorous approximation-error bounds for the neural-network parametrization lies beyond the present scope, we have added sensitivity experiments to network width, depth, and particle number to assess representation bias. revision: partial

standing simulated objections not resolved

A direct comparison against gold-standard function-space MCMC for the 2D Darcy-flow permeability inversion, which is computationally intractable at the problem size considered.

Circularity Check

0 steps flagged

No significant circularity; algorithmic proposal validated by experiments

full rationale

The paper proposes two algorithmic frameworks (FPI-BPINN and fParVI-PINN) for incorporating functional priors into Bayesian PINN inversion, using RFF to represent Gaussian priors, and evaluates them via numerical experiments on 1D traveltime tomography and 2D Darcy inversion. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central claims rest on empirical posterior estimation rather than tautological reductions. The methods are self-contained algorithmic contributions whose performance is assessed externally through simulation results.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 2 invented entities

The central claim rests on standard PDE forward modeling assumptions plus new algorithmic constructs whose representation power is not independently verified in the abstract.

free parameters (2)

Random Fourier feature hyperparameters
Chosen to represent Gaussian functional priors inside the neural network.
PINN architecture and training hyperparameters
Standard neural-network degrees of freedom that must be selected for both methods.

axioms (2)

domain assumption The governing PDE and boundary conditions are known exactly.
Required for any PDE-constrained inversion.
ad hoc to paper Neural networks with RFF can faithfully approximate the chosen functional prior.
Central to both FPI-BPINN and fParVI-PINN.

invented entities (2)

FPI-BPINN no independent evidence
purpose: Learn a weight-space prior that is consistent with a prescribed functional prior before performing Bayesian inference.
New algorithmic construct introduced by the paper.
fParVI-PINN no independent evidence
purpose: Perform particle-based variational inference directly in function space for PINNs.
New algorithmic construct introduced by the paper.

pith-pipeline@v0.9.0 · 5573 in / 1488 out tokens · 50490 ms · 2026-05-11T02:14:41.234424+00:00 · methodology

discussion (0)

Reference graph

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(2021)), and the proposed fpBPINN approaches, including FPI-BPINN and fParVI-PINN

18:Compute𝐠 𝑙 𝑖 = ∇ 𝜽𝑚(𝜽 𝑙 𝑢,𝑖,𝜽 𝑙 𝑚,𝑖)using Equation 38 19:end for 20:Update{𝜽 𝑙 𝑚,𝑖} 𝑛𝑝 𝑖=1 by one SGLD+R step using{𝐠𝑙 𝑖} 𝑛𝑝 𝑖=1 21:end for Agata and Okazaki:Preprint submitted to ElsevierPage 18 of 18 Functional-prior-based Bayesian PDE-constrained inversion using PINNs Algorithm 2fParVI-PINN Require:functional prior𝑝(𝐦), particle number𝑛 𝑝, evaluati...

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