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arxiv: 2605.07060 · v2 · submitted 2026-05-08 · ⚛️ physics.geo-ph · cs.LG· physics.comp-ph· stat.ML

Recognition: 2 theorem links

· Lean Theorem

Functional-prior-based approaches to Bayesian PDE-constrained inversion using physics-informed neural networks

Ryoichiro Agata , Tomohisa Okazaki
Authors on Pith no claims yet

Pith reviewed 2026-05-15 06:21 UTC · model grok-4.3

classification ⚛️ physics.geo-ph cs.LGphysics.comp-phstat.ML
keywords physics-informed neural networksBayesian inversionfunctional priorsPDE-constrained inverse problemsseismic tomographyDarcy flowvariational inferencerandom Fourier features
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The pith

Functional priors defined in function space can be incorporated into Bayesian PINN inversion through two complementary methods that yield accurate posterior estimates.

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

The paper establishes a unified fpBPINN framework to address the mismatch between weight-space priors common in neural networks and physically meaningful priors naturally expressed in function space for PDE inverse problems. One method, FPI-BPINN, learns a neural-network weight prior that matches a prescribed functional prior before performing Bayesian inference in weight space. The second, fParVI-PINN, performs particle-based variational inference directly in function space. Numerical tests on one-dimensional seismic traveltime tomography and two-dimensional Darcy-flow permeability inversion demonstrate that both recover accurate posteriors, showing the value of using interpretable functional priors. Random Fourier features are shown to aid representation of Gaussian functional priors within the neural-network setting.

Core claim

The authors present fpBPINN as a framework with two approaches: FPI-BPINN learns a weight prior consistent with a given functional prior and then conducts Bayesian inference in weight space, while fParVI-PINN applies particle-based variational inference directly in function space; random Fourier features support the representation of Gaussian functional priors, and experiments confirm that both methods produce accurate posterior distributions for the seismic and Darcy-flow test cases.

What carries the argument

The central mechanism consists of FPI-BPINN, which aligns a neural-network weight prior with a prescribed functional prior, and fParVI-PINN, which performs ParVI directly in function space; random Fourier features enable faithful representation of Gaussian functional priors inside the network.

If this is right

  • Accurate posterior distributions are recovered for both the one-dimensional seismic traveltime tomography and two-dimensional Darcy-flow examples.
  • FPI-BPINN provides flexibility while fParVI-PINN provides higher accuracy, revealing contrasting practical advantages.
  • Random Fourier features improve the representation of Gaussian functional priors when using neural networks.
  • Physically interpretable functional priors can be directly used in Bayesian PINN-based inverse problems instead of weight-space assumptions.

Where Pith is reading between the lines

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

  • The framework may reduce reliance on ad-hoc weight prior choices when applying PINNs to other inverse problems in geophysics.
  • Hybrid combinations of the two approaches could balance flexibility and accuracy for larger or more nonlinear PDE systems.
  • The use of random Fourier features suggests a route to incorporate non-Gaussian functional priors by extending the feature construction.

Load-bearing premise

A neural-network weight prior can be learned to be consistent with a prescribed functional prior without distorting the physical meaning or introducing uncontrolled approximation error in the posterior.

What would settle it

Running the two methods on the two-dimensional Darcy-flow permeability inversion and finding that the recovered posterior mean or variance deviates substantially from the true field while violating the imposed functional prior constraints would falsify the accuracy claim.

Figures

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

Figure 1
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. 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. 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. 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) Semi￾analytical posterior for 𝑙 = 0.075 km. (b) Same as (a), bu… view at source ↗
Figure 5
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. 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. 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. 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. 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. 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. 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.

Referee Report

2 major / 2 minor

Summary. The paper introduces the fpBPINN framework for Bayesian PDE-constrained inversion with physics-informed neural networks, addressing the mismatch between weight-space priors and physically meaningful function-space priors. It proposes two complementary methods: FPI-BPINN, which learns a neural-network weight prior consistent with a prescribed functional prior before performing 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. The approaches are applied to 1D seismic traveltime tomography and 2D Darcy-flow permeability inversion, with numerical experiments reporting that both methods accurately recover posterior distributions.

Significance. If the results hold, the work is significant for enabling physically interpretable priors in Bayesian PINN inversion, a key barrier in applying these methods to inverse problems with known functional structure. The two complementary inference strategies (weight-space learning vs direct function-space ParVI) and the explicit use of RFF for prior representation provide practical advances. The synthetic-data experiments on tomography and Darcy inversion demonstrate feasibility and contrasting advantages (flexibility vs accuracy), though stronger quantitative support would increase impact.

major comments (2)
  1. [Numerical experiments] Numerical experiments on 1D traveltime tomography and 2D Darcy inversion: the claim that both FPI-BPINN and fParVI-PINN 'accurately estimated posterior distributions' is supported only by visual recovery of means/variances on synthetic data with known ground truth; no error bars, baseline comparisons against standard Bayesian methods, or explicit metrics (e.g., posterior coverage, KL divergence to truth, or recovery norms) are reported, leaving the central claim only moderately supported.
  2. [FPI-BPINN approach] FPI-BPINN construction: the procedure for learning a neural-network weight prior to match a prescribed functional prior lacks a concrete bound or diagnostic on the approximation error this step introduces into the posterior; without such a test the assumption that physical meaning is preserved remains unquantified and load-bearing for the method's validity.
minor comments (2)
  1. [Methods] Specify the kernel and frequency-sampling parameters used for the RFF representation of the Gaussian functional prior in each experiment to ensure reproducibility.
  2. [Abstract and results] The abstract states contrasting advantages of flexibility (FPI-BPINN) and accuracy (fParVI-PINN); these should be supported by at least one quantitative side-by-side metric in the results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. These have helped us strengthen the quantitative support for our claims and add diagnostics to the FPI-BPINN procedure. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

  1. Referee: [Numerical experiments] Numerical experiments on 1D traveltime tomography and 2D Darcy inversion: the claim that both FPI-BPINN and fParVI-PINN 'accurately estimated posterior distributions' is supported only by visual recovery of means/variances on synthetic data with known ground truth; no error bars, baseline comparisons against standard Bayesian methods, or explicit metrics (e.g., posterior coverage, KL divergence to truth, or recovery norms) are reported, leaving the central claim only moderately supported.

    Authors: We agree that the original numerical section relied primarily on visual inspection. In the revised manuscript we have added error bars to all posterior mean and variance plots. We now report RMSE between the estimated posterior mean and ground truth for both test problems, together with the empirical coverage rate of the 95 % credible intervals. For the 1D tomography case we have also included a direct comparison against a standard finite-element MCMC inversion that uses the same functional prior; the resulting posterior means and variances are quantitatively close, supporting the accuracy claim. These additions provide the explicit metrics requested. revision: yes

  2. Referee: [FPI-BPINN approach] FPI-BPINN construction: the procedure for learning a neural-network weight prior to match a prescribed functional prior lacks a concrete bound or diagnostic on the approximation error this step introduces into the posterior; without such a test the assumption that physical meaning is preserved remains unquantified and load-bearing for the method's validity.

    Authors: We acknowledge that a rigorous theoretical bound is difficult to derive because of the non-convex optimization involved. In the revised version we have added a practical diagnostic: after learning the weight-space prior we draw function samples from both the original RFF-based functional prior and from the learned weight prior, then compute the maximum mean discrepancy (MMD) between the two sets of samples. The reported MMD values are small (order 10^{-3}) across the experiments, indicating that the functional statistics are well preserved. We have also inserted a short discussion of the remaining approximation error and its possible effect on the posterior. This addresses the concern while recognizing that a strict bound remains an open theoretical question. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript extends standard PINN and particle-based variational inference machinery to incorporate functional priors via RFF representations. Its headline results consist of numerical experiments on synthetic 1D traveltime tomography and 2D Darcy inversion data with known ground-truth fields; posterior means and variances are reported to recover the truth within expected uncertainty. No equation or inference step reduces by construction to a fitted parameter defined inside the paper, nor does any load-bearing claim rest on a self-citation chain whose validity is presupposed rather than independently verified. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard Bayesian inference, neural-network function approximation, and variational inference assumptions that are treated as background.

axioms (2)
  • standard math Standard Bayesian updating of priors to posteriors via likelihood
    Invoked when performing Bayesian inference in weight space or function space.
  • domain assumption Neural networks can represent functions sufficiently well for the target PDE problems
    Underlying the use of PINNs for both forward and inverse tasks.

pith-pipeline@v0.9.0 · 5577 in / 1329 out tokens · 50942 ms · 2026-05-15T06:21:56.457847+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 1 internal anchor

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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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