We propose a machine-learning algorithm for Bayesian inverse problems in the function-space regime. Based on one-step generative transport, the method learns an amortized neural operator whose pushforward of a Gaussian source approximates the posterior distribution conditioned on each new observation. We show that white-noise sources are incompatible with the function-space limit, and therefore adopt a prior-aligned GRF as the source. We justify this choice through the Lipschitz regularity of the resulting one-step conditional posterior transport and numerical experiments on linear inverse and PDE-based inverse problems. The method is not distilled from MCMC: it is trained only with prior samples and simulated partial noisy observations. Once trained, it generates a 64×64 posterior sample in ∼10−3s, avoiding repeated forward-model evaluations in MCMC and repeated network evaluations in multistep generative samplers while matching key posterior summaries.
Figures & tables
Figure 1: Conditional neural-operator parameterization of the mean-flow predictor. A source draw ξ∼ρ is lifted by P , passed through L conditional operator layers, and projected by Q to produce the mean-flow correction. During training, time pairs 0≤r≤t≤1 are sampled randomly; at inference, the learned predictor is evaluated at (r,t)=(0,1) to generate one-step posterior samples.
Figure 2: Linear inverse problem. The first row shows the input field, sparse observations, reference posterior mean, predicted posterior mean, and mean error. The second row shows representative posterior samples generated by the proposed sampler.
Figure 3: Energy spectra E(k) of the posterior distributions for the 65×65 linear inverse problem under different choices of source kernels.
Source
ϵμeval
ϵσeval
ρC
4.94×10−2
2.72×10−2
ρW
8.24×10−2
4.13×10−2
ρ1
6.28×10−2
2.49×10−2
ρ2
8.15×10−2
8.33×10−2
Table 1: Effect of the source measure on posterior approximation for the linear inverse problem.
Figure 4: Darcy inverse problem. The first row shows the unknown field, sparse observations, reference posterior mean, predicted posterior mean, and mean error. The second row shows representative posterior samples generated by the proposed sampler.
Figure 5: Advection inverse problem. The first row shows the unknown field, sparse observations, reference posterior mean, predicted posterior mean, and mean error. The second row shows representative posterior samples generated by the proposed sampler.
Figure 6: Reaction-diffusion inverse problem. The first row shows the unknown field, sparse observations, reference posterior mean, predicted posterior mean, and mean error. The second row shows representative posterior samples generated by the proposed sampler.
Figure 7: Navier-Stokes inverse problem. The first row shows the unknown field, sparse observations, reference posterior mean, predicted posterior mean, and mean error. The second row shows representative posterior samples generated by the proposed sampler.
Sampler
Steps
Sampling time (s/sample)
MCMC (Darcy)
3×104
2.05×101
MCMC (Advection)
3×104
7.01
MCMC (Reaction–Diffusion)
3×104
1.04×101
MCMC (Navier–Stokes)
3×104
1.97×101
Multistep diffusion
320
1.05×10−1
Multistep flow matching
320
1.67×10−1
Table 2: Wall-clock generation time for producing posterior samples conditioned on a single fixed observation yobs . Hardware: MCMC is run in parallel on 16 cores of the CPU ( 32 thread at 3.6 GHz) cores; learned samplers are evaluated on a single NVIDIA RTX A6000 GPU. The per-sample clock time is computed by averaging over the generation time of 1000 posterior samples.
Dataset
Source
ϵμeval
ϵσeval
Darcy
Anisotropic
5.55×10−2
4.27×10−2
White
8.26×10−2
4.30×10−2
Advection
Anisotropic
6.08×10−2
5.09×10−2
White
1.63×10−1
5.14×10−2
Reaction– Diffusion
Anisotropic
7.62×10−2
8.68×10−2
White
7.89×10−2
8.91×10−2
Table 3: Posterior approximation errors of the conditional one-step sampler on PDE inverse problems.
Appendix figures & tables9 assets
Supplementary material from the paper’s appendix.
Appendix
Dataset
Source
ϵμeval
ϵσeval
Darcy
Anisotropic
5.55×10−2
5.45×10−2
White
7.56×10−2
4.37×10−2
Advection
Anisotropic
6.36×10−2
3.76×10−2
White
1.43×10−1
5.08×10−2
Reaction–Diffusion
Anisotropic
7.23×10−2
7.94×10−2
White
8.08×10−2
7.85×10−2
Appendix
Table 4: Posterior approximation errors of the PODNO-parameterized Mean-Flow sampler.
Dataset
Source
ϵμeval
ϵσeval
Darcy
Anisotropic
1.02×10−1
1.48×10−1
White
1.41×10−1
1.02×10−1
Advection
Anisotropic
7.34×10−2
1.37×10−1
White
2.04×10−1
1.04×10−1
Reaction–Diffusion
Anisotropic
9.17×10−2
2.04×10−1
White
1.44×10−1
1.44×10−1
Appendix
Table 5: Posterior approximation errors of the U-Net-parameterized Mean-Flow sampler.
Dataset
Source
ϵμeval
ϵσeval
Darcy
Anisotropic
5.43×10−2
6.16×10−2
White
2.80×109
3.14×109
Advection
Anisotropic
7.19×10−2
5.05×10−2
White
1.11×1010
1.91×105
Reaction-Diffusion
Anisotropic
8.23×10−2
9.64×10−2
White
2.16×1010
6.14×105
Appendix
Table 6: Posterior approximation errors of the improved Mean-Flow sampler.
Dataset
Source
ϵμeval
ϵσeval
Darcy
Anisotropic
1.08×10−1
7.00×10−2
White
1.19×10−1
7.71×10−2
Advection
Anisotropic
9.90×10−2
4.30×10−2
White
2.99×10−1
8.24×10−2
Reaction-Diffusion
Anisotropic
1.08×10−1
1.02×10−1
White
1.16×10−1
9.21×10−2
Appendix
Table 7: Posterior approximation errors of the multistep diffusion baseline.
Figure 8: Effect of the number of test-time sampling steps on posterior accuracy for the linear inverse problem for the multistep diffusion model. We use the same model trained with 320 sampling steps and vary only the number of sampling steps at inference time.
Figure 9: Comparison of posterior statistics from our model and DPS inference on the same 65×65 linear test instance as in Figure 2 . DPS uses an unconditionally trained diffusion model with a guidance weight, and we report the best-performing result obtained by tuning this weight over [0,1] , attained at 0.1 .
Figure 10: Comparison of posterior statistics from our model and DPS inference on the same 65×65 darcy test instance as in Figure 4 . DPS uses an unconditionally trained diffusion model with a guidance weight, and we report the best-performing result obtained by tuning this weight over [0,1] , attained at 0.1 .
Symbol
Description
Ω
Spatial domain
d
Spatial dimension
s
Sobolev regularity exponent defining the distribution space H−s
X,x,X
Unknown random function, its realization, and encoded variable
Yobs,yobs,Yobs
Observation random variable, its realization, and encoded observation
Ξ,ξ
Source random variable and its realization, sampled from ρ
Appendix
Table 8: Notation.
Setting
Symbol
Value
FNO Fourier modes
K
12
FNO layers
L
4
Network width
Nwidth
96
Learning rate
lr
10−4
Flow ratio
νflow
0.25
Batch size
Nbatch
20
Appendix
Table 9: Network architecture and training hyperparameters.
We consider amortized Bayesian inference for nonlinear inverse problems in settings where only samples from the joint distribution of parameters and observations are available. Classical methods such as Markov chain Monte Carlo require solving a new inference problem for each observation, which can be computationally prohibitive when inference must be repeated many times. We propose a transport-based approach that learns an observation-dependent map pushing forward a reference measure to approximate the posterior distribution. The map is trained by minimizing an averaged energy-distance objective between the true posterior and the learned pushforward. This formulation is likelihood-free, requiring only joint samples, and avoids density evaluation, invertibility constraints, and Jacobian determinant computations. For function-space inverse problems with Gaussian priors, we parameterize the transport map as the identity plus a perturbation in the Cameron-Martin space of the prior, preserving absolute continuity with respect to the prior. In infinite-dimensional settings, the map is represented using neural operators. We illustrate the method on a finite-dimensional nonlinear inverse problem and two PDE-constrained inverse problems arising in porous medium flow and seismic inversion. The results show that the learned transport captures posterior structure, including multimodality and dominant modes, while enabling fast posterior sampling for new observations.
Hojjat Kaveh, Ricardo Baptista, Andrew M. Stuart
California Institute of Technology · University of Toronto
This paper addresses the issue of inversion in cases where (1) the observation system is modeled by a linear transformation and additive error, (2) the problem is ill-posed and regularization relies on a Bayesian strategy, (3)~the prior is modeled by a diffusion process adjusted on an available large set of examples. In this context, it is known that the issue of posterior sampling is a thorny one and the paper introduces a Gibbs algorithm. It appears that this avenue has not been explored, and we show that it is particularly effective and remarkably simple. In addition, it provides clear elements regarding convergence guarantees in a specific case and arguments supporting such guarantees in practical cases. The results are clearly confirmed by numerical simulations based on a toy example.
Jean-François Giovannelli
Groupe Signal-Image, IMS (Univ. Bordeaux, CNRS, BINP), Talence, France
Principled regression for stochastic processes is a long-standing challenge with deep connections to scientific inverse problems. We introduce Flow Annealing Posterior Sampling (FLAPS), to our knowledge the first function-space posterior sampling framework that unifies stochastic-process regression and PDE inverse problems. Built on pretrained function-space flow-matching priors, FLAPS enables likelihood-guided posterior inference from sparse and noisy observations, supports variable query discretizations, and avoids explicit prior-density evaluation. Its Langevin correction uses a low-rank covariance preconditioner to exploit dominant function-space correlations across discretizations. Across Gaussian and non-Gaussian stochastic-process regression benchmarks and diverse PDE inverse problems, FLAPS produces coherent posterior samples with well-calibrated uncertainty quantification, significantly outperforming existing functional regression baselines and achieving competitive or better noisy PDE inverse performance than diffusion-based posterior samplers while reducing test-time sampling cost.