Geometric Autoencoder Priors for Bayesian Inversion: Learn First Observe Later
Authors: Arnaud Vadeboncoeur
Gregory Duthé
Mark Girolami
Eleni Chatzi
Venue: ICLR 2026 Poster (2026)
Keywords: Bayesian Inverse Problems
Physics
Graph Machine Learning
Generative Modelling
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: We introduce Geometric Autoencoders for Bayesian Inversion (GABI), a framework for learning geometry-aware generative models of physical responses that serve as highly informative geometry-conditioned priors for Bayesian inversion.

Abstract: Uncertainty Quantification (UQ) is paramount for inference in engineering. A common inference task is to recover full-field information of physical systems from a small number of noisy observations, a usually highly ill-posed problem. Sharing information from multiple distinct yet related physical systems can alleviate this ill-posedness. Critically, engineering systems often have complicated variable geometries prohibiting the use of standard multi-system Bayesian UQ. In this work, we introduce Geometric Autoencoders for Bayesian Inversion (GABI), a framework for learning geometry-aware generative models of physical responses that serve as highly informative geometry-conditioned priors for Bayesian inversion. Following a ''learn first, observe later'' paradigm, GABI distills information from large datasets of systems with varying geometries, without requiring knowledge of governing PDEs, boundary conditions, or observation processes, into a rich latent prior. At inference time, this prior is seamlessly combined with the likelihood of a specific observation process, yielding a geometry-adapted posterior distribution. Our proposed framework is architecture-agnostic. A creative use of Approximate Bayesian Computation (ABC) sampling yields an efficient implementation that utilizes modern GPU hardware. We test our method on: steady-state heat over rectangular domains; Reynolds-Averaged Navier-Stokes (RANS) flow around airfoils; Helmholtz resonance and source localization on 3D car bodies; RANS airflow over terrain. We find: the predictive accuracy to be comparable to deterministic supervised learning approaches in the restricted setting where supervised learning is applicable; UQ to be well calibrated and robust on challenging problems with complex geometries.

Published as a conference paper at ICLR 2026
GEOMETRIC AUTOENCODER PRIORS FOR BAYESIAN INVERSION: LEARN FIRST OBSERVE LATER
Arnaud Vadeboncoeur"
Gregory Duthé
Department of Engineering
Institute of Structural Engineering
University of Cambridge Cambridge, 7a JJ Thomson Ave, UK av537e cam. ac.uk
ETH Zürich
Zürich,
5, Switzerland duthe@ibk baug. ethe.
Mark Girolami
Eleni Chatzi
Department of Engineering University of Cambridge Cambridge, 7a JJ Thomson Ave, UK mag920 cam. ackuk
Institute of Structural Engineering
ETH Zürich
Zürich, Stefano-Franscini- Frano-sninnnii-altaaa 5, Switzerland
chat 1ibk. baug. ethz.
ABSTRACT
Uncertainty Quantification (UQ) is paramount for inference in engineering. A common inference task is to recover full-field information of physical systems from a small number of noisy observations, a usually highly ill-posed problem. Sharing information from multiple distinct yet related physical systems can alleviate this ill-posedness. Critically, engineering systems often have complicated variable geometries prohibiting the use of standard multi-system Bayesian UQ. In this work, we introduce Geometric Autoencoders for Bayesian Inversion (GABI), a frame- work for learning geometry-aware generative models of physical responses that serve as highly informative geometry-conditioned priors for Bayesian inversion. Following "learn first, observe later" paradigm, GABI distills information from large datasets of systems with varying geometries, without requiring knowledge of governing PDEs, boundary conditions, or observation processes, into a rich latent prior. At inference time, this prior is seamlessly combined with the likelihood of specific observation process, yielding a geometry-adapter posterior distribution. Our proposed framework is architecture-agnosiic A creative of Approximate Bayesian Computation (ABC) sampling yields an efficient implementation that utilizes modern GPU hardware. We test method steady-state heat over rect- angular domains; Reynolds Reynolds-Averaged Helmholtz resonance and source localization 3D car bodies; RANS airflow over terrain. We find: the predictive accuracy to be comparable to deterministic supervised learning approaches in the restricted setting where supervised learning is applicable; UQ to be well calibrated and robust on challenging problems with complex geometries.
Stoier-Stokes (RANS) flow around airfoils;
INTRODUCTION
Many important problems in engineering deal with spatially varying quantities that must be inferred. The task of inferring full-field information from sparse noisy measurement type of inverse problem (Kaipio & Somersalo 2005); these often need to be regularized due to their inherent ill-posedness (Calvetti & Reichel 2003). Statistically interpretable regularization can be achieved through the Bayesian paradigm (Stuart 2010)), where we combine likelihood function characterizing data fit for particular observation process, and a prior which characterizes assumptions about the data- generating process. Bayesian inference allows for robust and principled uncertainty quantification. where we obtain a distribution on the estimated quantity of interest given data from a particular physical system. The main challenge in enacting Bayesian program is prior specification; often, not
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much can be said about the prior which often leads to diffuse and vague posteriors with relatively poor predictive accuracy when compared to deterministic methods. To alleviate this vagueness, one possible strategy is to learn suitable prior from dataset of related but distinct physical setups. where more data may be available. This is explored in general-purpon
Al applications (Shwartz-Ziy
et al. 2022 Feng al.; 2023] [Boss al., (024) and physical applications (Meng et al. 2022 Akyildiz et al. [2025] Patel & Oberai, 2021] Patel et al., [2022] Laloy et al. 2018 2017). However, in
engineering, geometry plays a crucial role in determining the dynamics arising for physical system. Hence, for truly general purpose, practical methodology. learned priors must incorporate knowledge about the geometry of the problem at hand. But changing geometries hinder the direct learning of a prior over fields as the probability spaces involved are tied to the geometry of each individual physical system, prohibiting a direct treatment.
In this work, we propose methodology using graph-based autoencoders to learn highly informative priors as geometry-aware generative models for use in Bayesian inversion tasks. We also show how to effectively solve the Bayesian inverse problems resulting from these priors through a pushforward model. Importantly, the proposed scheme decouples the task of "learning" and the task of "inferring from observations". resulting in train-once use-everywhere approach which is independent of the particular observation process for solving inverse problems, a trait not shared by related supervised learning approaches. We make the following contributions:
(C1) We propose a methodology to encode geometries and physical fields into a learned latent prior without specifying the governing PDE or boundary conditions.
(C2) We prove that solving the Bayesian inverse problem in the learned latent representation solves the same problem as the inversion in the original space with pushforward prior.
(C3) The proposed methodology is naturally extended to the Bayesian estimation of other impor- tant quantities in the observation process, such as learning the observational noise.
(C4) We test the proposed methodology on: a steady-state heat problem, RANS flow around airfoils of varying geometries, damped Helmholtz resonance and source localization on geometries. We compare our methods to supervised neural network-basee methods and Graph Gaussian Processes where relevant.
(C5) We demonstrate the scalability of the method on a large terrain flow problem through a multi-GPU implementation, showing potential for training GABI foundation models.
Section discusses the inference setup in detail; Section 1.2 goes over relevant related works;
Section deseribes the proposed methodology: Section discusses the implementation of the proposed method; fection[Jtestt and compares the proposed method; Appendices | C elaborate on proofs, additional numerical results and implementation details, relevant discussions on VABs.
1.1 SETUP
Let un E Un := U( U(,,;; be (possibly vector valued) function representing the full field behavior of physical system on the bounded domain Mn C Rdd. The space U is the relevant function space for the full field solution, e.g. a Sobolev space, and the geometric domain M, may be a manifold with boundary or a boundaryless closed manifold. In this work, un will be: heat on surface, fluid flow in domain, and resonance patterns on an object. Here, the index n refers to the different physical systems which possibly have different geometries, forcing, and boundary conditions. In numerically representing these continuous systems, we obtain discretized node values un E Un := U(M,; Rdd)) which live on M, = (Vn. E,) where V,, = {aI..... Td.. C M, is set of vertices and E,, is set of edges connecting the vertices. As such, we represent M,, as an undirected graph describing the physical geometries on which the physics is taking place, i.e., computational mesh. We assume to have a dataset D {u,, M,, J.y [] This dataset can be constructed from simulation data, as is the case in this paper, or from full-field data from experiment such as through particle image velocimetry, digital image correlation, magnetic resonance, electrical impedance tomography, infrared thermography etc.
First, in the proposed methodology we learn a Mn-deppendnt prior generative model for u.. Second, using this learned highly informative prior and sparse noisy observations, yo E dd. of new
N., M., J. can be viewed as a dataset (u,, Ms | indexed by dummy variable such that maps to (i, n).
A dataset with multiple full-fields per geometry M.. as [[u,,i)
When discussing observations which are associated to geometry not in the dataset D we change the index from o for clarity of exposition.
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physical system new geometry, Mo: solve the Bayesian inverse problem to obtain the posterior distribution Pu.ly.' We assume to know the discretized geometry M, associated to yo. In this paper, the observations and discretized solution field are related by:
Yo = H. + E.: &. (0,,
(I)
We take H. : Idd + Rd to be matrix of zeros and ones selecting the nodes to be observed. This assumed to be known observation operator corresponding to sensor placements. We note that many observation models can be considered with ease, such as surface integrals for computing drag coefficients, Radon transforms for tomography, as well as any number of noise models additive or not. As our learned prior is entirely independent of the observation process, changes to the observation process do not necessitate retraining and can be incorporated "on-the-fly".
The proposed methodology works follows:
In the training stage, using a dataset of geometries and full-field solutions on these ge- ometrics, D {u, J we learn a graph-based autoencoder which embeds the joint distribution of solutions and geometries into latent Gaussian distribution. This will act as our highly informative geometry-awan
data-driven prior.
In the application stage, we use this learned prior solve inverse problems for data coming from new physical systems not in the dataset D. That is, given sparse and noisy observations Yo of solution u, of physical system with geometry M.. we use our previously learned prior to solve the Bayesian inverse problem for the unknown full-field u. We recover a distribution over the solution u, by only observing Yo.
Using the proposed methodology, one can encode prior beliefs through geometry-dependeet full-field solution datasets. Practitioners can then construct highly-infone informative priors from data across varying physical geometries. This prior is independent of the observation process for any particular inference task, hence this is learn-once once eeply-everywhere approach.
1.2
RELATED WORKS
1.2.1 DIRECT MAP INVERSION
Many interesting approaches have been recently proposed which directly learn an amortized map from observable solution/parameter. These be subdivided into deterministic methods and probabilistic often Bayesian approaches. Most related to the setting considered in this paper the work in Duthé al.] (2025). Here, the authors consider a graph-ML approach to learning deterministic map from observable yo to the underlying vector field u,, The supervised methodology used in the comparisons in this work draws inspiration from that prior work. Another strongly related methodology can be found in Arridge et al. (2019) although here the problemobeemmm of changing geometry
not directly tackled. Other noteworthy works in this area include (Adler & Öktem 2017 Dittmer
et al. 2020). Interesting works look at probabilistic schemes for constructing conditional maps from
observables to quantities of interest (Kaltenbach et al.) [203]] Baptista et al.) [024]] Hosseini et al.
2025). Other works in this area include (Adler et al. 2022 Ardizzone et al. 2018).
1.2.2 FORWARD AND INVERSE PROBLEMS ON GRAPHS
Graphs are handled in machine learning frameworks through geometric deep learning (Bronstein et al. 2 Wu et al. 2022 Kipi 2016). Several important works consider inverse problems where the quantity inferred lives of a graph description of the geometry. In Garcia Trillos & Sanz-Alonss (2018) the authors prove the convergence of the graph posterior to the continuum posterior subject to some conditions. In PPovaaa et al. 2022) the adjacency structure of a mesh is leveraged to accelerate variational inference inversion. Important works consider solving the PDE forward problem through physics-informed informed graph machine learning(Sco & Liu 2019 Valencia et al. 2025 Wandel et al. 2020) Used in our comparisons are linear inverse problems with Gaussian process priors, this leads to graph Gaussian process methods (Borovitskiy et al., 2021] Mostowsky et al. 2024). We find it important to mention works in the physics-informed literature where (assuming complete or partial knowledge on PDEs, boundary conditions, forcing etc.) one can use physics residuals to learn forward and inverse maps for PDEs (Raissi et al. 2019: Gao et al., 2022; Vadeboncoeur et al., 2023 Xu et al. 2023 Rixner & Koutsourelakis 2021). However, in this work, we focus solely on methods which do not assume knowledge of the governing PDE, boundary conditions, or forcing and are entirely data-driven in their approach.
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1.2.3 STATISTICAL AUTOENCODERS
Related to inverse problem are latent variable representations of graph data structures. In this category, we focus on the seminal work of the graph VAE (Kipl' & Welling 2016 Nazari et al. 2023). We refer the reader to Appendix for a more elaborate discussion on the relationship between VAEs and our proposed framework. Related to this are the works (Mylonas et al.
2021 Liu et al.
2019). The
proposed work centres around a class of autoencoders dubbed statistical autoencoders, which there are many variants (Zhao et al. [019 Kolouri et al.) 2019 [Turniiii 2021: Tolstikhin et al.
2018). We
specifically focus on the MMD (Gretton et al. (012) variant of the autoencoder. Many other variants could be considered. Another relevant class of autoencoders are conditional autoencoders (Sohn
et al. 2013), we will make use of related ideas by conditioning encoder and decoder on geometry.
1.3 NOTATION
We denote by P. a probability measure on the measurable space (2,S), and by pz the associated density with respect to Lebesgue measure when it exists. Empirical measures are mixture of Dirac measures. That is, for D {z, ] 1 if E A, and 6z, (A) = 0, if z; & A. We will sometimes refer to such probability measures as the "densities" p.q lighten notation. Superscripts in parentheses denote realizations of random variables. For a function
have [ 1/4 Ei=1 8z, where 82, (A)
M
- U, the pushforward I#Ps means I& (A) = TP (g (A)) for all
measurable sets A. We note that sample (m)) ~ P= is obtained as (n) g(2(m)), 2 ~ Ps.
2 METHODOLOGG
We now describe the proposed methodology in detail. In Section 2.1 we carefully lay out the central result allowing for the Bayesian treatment the inference task; that is, the equivalence between the Bayesian posterior of pushforward prior (our generative model), and the Bayesian inverse problem in the implied latent space of the pushforward prior. Following from this result, in Section 2.2 we describe our two-step procedure: first, learn a generative model to be used Bayesian prior; second, observe sparse data for the inverse problem and solve the Bayesian inverse problem in the latent space push this posterior back out to the desired physical field living on the geometry of interest. In Section 2.3 we show how to extend our methodology to estimate the noise in the observations.
Geometry-conditiong
2.1 POSTERIOR FROM PUSHFORWARD PRIOR
We present the lemma motivating our methodology; the equivalence between the Bayesian posterior with pushforward latent prior and the pushforward of the Bayesian posterior over the latent space.
Lemma 2.1. Let
and (Z,S,Ps) be measure spaces and g : Z U be (S,F) measurable. Furthermore, let P Puly be the Bayesian posterior on U with likelihood proportional to exp(-4(u;g)) y)) and prior F. == 9 P.. We find
P Puly = 9 IP zly-
where dIP: (1)=)y()) 1/Z(y) exp(- (g(()) g) (z) is the Bayesian posterior on Z.
Stated simply, in Bayesian inference, when a prior on 14 is specified as the pushforward under (Subsection I.3) of latent prior space 2, two distributions equivalent: i) the Bayesian posterior on UT and ii) the pushforward under of the posterior associated to the latent prior on Z. This result will be explicitly related to the geometric inference setting in Theorem 2.2 showing that with latent prior constructed via geometry-awan encoder/decoder, we can solve inverse problems on 2/ with single common latent representation relevant any number of in-distribution geometries. All proofs can be found in Appendix A
2.2 GEOMETRIC AUTOENCODER PRIORS FOR BAYESIAN INVERSION (GABI)
We describe the two-step process of learning a coanetry-awars prior and how to use this to solve inverse problems.
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Step 1: Learn Geometric Autoencoder Prior. Let Z Rd be latent vector. We pose the -eedeenneent maps
E Un - -, where E (un) = E. (un: Mn); D + Un where (z) := (z; M.).
(2a)
(2b)
Both Ex (the encoder) and DV (the decoder) are graph neural networks with parameter sets 0, respectively. These two maps imply geometry conditioned autoencoder. We use Ep|-] as shorthand for N (uun,M))eoll Set the parameters of the auto-encoder to
,0 arg min Ep[l|u, - (D (u,)) d(p2:
8,
where p = Ep (x,) is the empirical distribution of encoded solutions, 02 = (0, I), arg min to approximated with gradient descent, and d P(Z) x P(Z) - [0, oo) may be any suitable divergence that is well-posed for comparing empirical measures: maximum mean discrepancy (MMD), energy distance (ED), Wasserstein, or Wasssersssn distances etc. In this work, we focus MMD.
Step 2:
Observe and Sample Posterior. For new data yo associated to geometry Mo. we re-write the data generating model in (T) in terms of a decoded latent variable
Yo H (2) E.. E. ~ ~N0,
where DØ - ( M.). We specify our prior distribution over u., denoted Pa,, using the trained decoder; and specify the likelihood, denoted py, associated this decoded observation model (4
P
D o# 92i
P yul= ((),
(5)
This likelihood along with prior tu imply a posterior
Py (Yo) (z(z)
Px yo (z)
(6)
Py. (yo)
Theorem 2.2. Assuming the relevant measurability of D the posterior
Pully. = D DD Paly.
(7)
The posterior is sampled by first sampling from Pxy. in (6 and decoding with DE Hence, by first solving the inverse problem in the latent space, we can use Theorem 2.2 to easily obtain posterior samples in the decoded solution space for given geometry. This allows us learn unified prior over the solution field generative model over varying geometries and use to solve Bayesian inverse problems with a highly informative learned prior on new geometries.
2.3 OBSERVATIONAL NOISE ESTIMATION
Beyond the benefits of quantifying uncertainty related to model predictions, the Bayesian paradigm offers a principled framework for estimating other aspects of the data-generating model. We highlight that such estimations are completely independent of the training of the autoencoder prior this is not the case for direct regression methods which need to have all observation variables incorporated training time. We now consider the common task of jointly estimating the observational noise standard deviation (now indexed by as it may vary from one system to another), together with field of interest u. This achieved through the latent joint posterior
P Volla (yo)Pa.o (z, T.)
D P2 (2,
Py.(y.. (Yo)
where Pr.o,(z,03, qzl (o, (oo) for a specified prior Po. Po. (oo).
Corollary 2.3. We can sample the joint posterior the solution fields through the pushforward P Pu..ooly. (D) @ Id.) # p P2...ly.' where Id. is the identity map.
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Algorithm I: Inversion with GABI-ABC
Specify the observation vector yo. discretized geometry Mo. total sample budget N,, number of accepted samples Na.
2: Train E., D through (3 on D.
3: for [PatcchedPPaaalelllli
i == 1: N, do
4: Sample (4) Qz
5: Decode u (8) D (z Mo)
6: Apply observation process with noise: yo (8) Ho (0)
:0 -0 ~(~,oo,)')))
7: Compute r; ly yo ll.
8: return return N₂ samples of uo (1:N5) with least residual ri-
3 IMPLEMENTATION
Neural Networks: We make use of meshes and graph neural networks to describe and manipulate geometric information. The proposed methodology is agnostic to any particular architecture, however we need the following characteristics: Geometry-aware architecture and sata-struuctur; Mapping to and from fixed peed-dimensionll vectors; Non-locality. We expand on these points in B For the heat, airfoil, and car problems we make use of Graph Convolutional Network with interleaved nonlocal averaging layers (Lanthaler et al. 2024). For the terrain flow problem we use GEN (Li et al. (0200) GNN architecture. We detail these architectures Appendix B.I We also test a Transformer on the heat problem in Appendix B.3.3
Sampling:
To realize GABI, we first train the autoencoder; then, we sample from the corresponding latent posterior in (6) and decode the resulting latent samples to obtain samples from the posterior for a given geometry. We note two alternatives for sampling from the posterior: MCMC, and ABC sampling. We find ABC sampling to be advantageous for this task due to: (i) the massive parallelization capabilities of neural networks as opposed to the sequential nature of MCMC; (ii) the specification of a well-tailored prior distribution the prior is not unnecessarily vague leading to good overlap between the prior and posterior; (iii) the low dimensionality of the observational data removing the need to use possibly problematic summary statistics. In Algorithm we outline the implementation of the GABI-ABC variant. For MCMC sampling, we use the NUTS algorithm (Hoffmma et al. 2014
Notes on Alternative Approaches: Physics Driven Inversion. These require additional knowledge aside from access to a representative dataset and information specific each system we wish to query such as the governing PDE, boundary conditions, forcing etc. VAEs. We refer the reader to Appendix for an in-depth discussion on the unsuitability of VAES for the task we are solving, both in terms of the overall methodology and in terms of the autoencoder used within the proposed methodology. Supervised learning/Direct Map Inversion. One can choose to Iearn a supervised learning direct mapping (y,, M,) + u, this approach is the main point of comparison with our method. The significant disadvantage of this approach is that the observation process (in this paper y, H u, E..) must be known at training time and kept the same at inference time (or kept the same in distributional sense, as explained in the comparisons). However, in practice, the observation process for different physical systems can vary enormously as we measure different quantities such as moments, local averages, summary coefficients, have a variable number of observations, or vary the type and magnitude of noise etc. In contrast this, GABI independent of the observation process at training time and only requires this information at test time. Thus, GABI offers a train-once-use everywhere type foundation model, hugely advantageous for practical/industry uptake of ML models. Gaussian Process Regression. For an inverse problem over fields, natural choice of prior is Gaussian process prior. As the observation process H, linear, the classical Bayesian posterior for this problem is precisely Gaussian process regression (GPs). We perform hyper-parameter optimization of this GP, which relates it to common usage of GPs in the context of ML. The GPs are defined directly on the graphs (Borovitskiy al. 2021). Bayesian NNs. The problem of specifying informative priors over varying geometries persists and inherits the problems of direct maps discussed above.

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44
(a)
(b)
Figure 1: Four out of 1k geometries and solutions in dataset for the steady- heat problem.
0.303
0.225
0.2
40-
011
00075
20-
0.0
0000
Profleeted
(a) Selected query locations
(b) marginal of Pu,, bra specified locations
Figure 2: (a) Four selected query locations for the sampled predictive solutions given data, the full field estimations are in Figure (b) The corresponding histograms for the predictive posterior at specified query locations; the dashed black line indicates the ground truth for each histogram.
NUMERICS
In this section we present numerical results for steady-state heat on rectangles, flow fields around airfoils, damped resonance and source localization on car bodies, and terrain flows. We compare our method to direct regression (Direct Map) using graph NNs, and Gaussian process regression with Matérn kernels 3/2, and radial basis function (RBF). In all experiments training time (Train) on Nvidia RTX 4090 GPUs (the terrain problem is multi-GPU), and the mean per-geometry prediction time (Pred.). We also note that in using ABC for sampling from the posterior we do not need to compute the likelihood function. This may have strong advantages in application where the observation process yields intractable likelihoods. In using GABI with MCMC, as in the comparisons, we do compute the likelihood. In training the latent prior for GABI, the main way we have of knowing if the autoencoder networks perform adequately for the inference task, is by assessing the quality of reconstruction for a holdout set, as well as how close the embed holdout set being Gaussian the latent space, which assessed qualitatively and quantitatively.
report the wall-clock
4.1 RECTANGLES STEADY-STAVE HEAT
In this section, we study the homogeneous steady-state heat equation on rectangular domains with varying width, height, and boundary conditions. Thus, for a given rectangle M,, boundary condition hys any un satisfies the steady-state heat equation. We take M,, C R2 to M,, {ar R w, The domain M, is defined by its length and width (I,, wx) drawn
0 21 < 4n,0 0 < from uniform(( (0.oooo0011.0.0000 0.1,,..)...).),, (1, 1)); we specify ht, to be zero on the bottom and left side, and drawn from unifo 000fformm(11., 0), (1)
the top and right. The solution to ( depends on Mn, An. We generate 1k solutions to be in the training dataset. In Figure I we show example geometries and solutions in the dataset. Once the model is trained on this data (the loss is shown in (B.)) we select new observation vector and geometry M..y. and use ABC sampling as in Algorithm I Figure [ (b) shows 4 coloured histograms corresponding the marginal distribution of decoded posterior
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Table I: Comparison on Heat Equation in Rectangular Domain
Method
Field
MAE
% 1 std
% std
Train
Pred.
2D Heat Rect. Domain
GABI-ABC
1.58 10 1.36
10-2
80.91% 66.66%
95.59%
2.62hr
0.908s
GABI-NUTS
111--00241.000
10-2
96.18%
"
410.31s
Direct Map
1.25-10-22. 1 -10-2
1.47hr
0.0029I
GP (M 1/2)
146--102244 1002
66.36%
89,45%
0.65s
GP (M 3/2)
17
8.06 100-10022- 1 4.85
10-2
65.80%
87.22%
0.64s
GP (RBF)
1.39 10 6.66
10-2
14.88%
29.36%
0.64s
Table Comparison on Heat Equation in Rectangular Domain Unkown Noise
Method
Field/Qol
MAE
% 1 std
% std
Train
Pred.
GABI-ABC
(u)
1.09-10-21- -
75.74%
94.76%
2.62hr
0.904s
Direct Map
(2)
113--10-22..7-100 64.87%
1.46hr
0,0030s
GP (M 1/2)
87.50%
0.615s
GP (M 3/2)
(u)
1.56-10=45.56-11
51.58%
76.94%
0.686s
GP (RBF)
(u)
140-110'16.94--
18.03%
36.69%
0.694s
GABI-ABC
(g) 7.96 100 5.74 10
42.10%
69.11%
Direct Map
(0)
GP (M 1/2)
(
1.08 1 + 1.03 10
GP (M 3/2)
(
1.10
1.47
GP (RBF)
2.01 10 33.1
(based on 10 sparse observations in the vector yo) at the 4 queried locations on Figure (a). As the orange dot is close to a boundary with value the entire dataset (4 representative samples from the Ik geometry dataset can be seen in Figure [T. GABI presents very small uncertainty in this area. As we query points areas where there is more variability of the solution in the dataset, GABI presents wider uncertainty. In all cases the mean of the GABI posterior is very close the ground truth (vertical dashed line). This exemplifies the balance GABI strikes between sparse data and informative data-driven priors. Comparison Known Noise. In Table I we compare two sampling variants of GABI with a direct regression map as well as Gaussian Process Kriging with various kernel M stands for Matérn. In this setup, we keep the number of observations at 10 randomly selected locations and the noise standard deviation at 10 10-2 Comparison Unknown Noise. In Table 2 we test the noise estimation capabilities of GABI. We have the noise standard deviation be random and drawn from shifted lor-normal distribution as a, = exp(s, 4) + 10-3;e, ~ NO, 7). We estimate the noise for each data yo with the GABI approach outlined in 2.3
4.2 AIRFOILSS FLOW FIELD
In this section, test the proposed methodology the full field reconstruction airflow around an airfoil from a dataset of Reynolds averaged Navier-Stokes simulations. We use the setup described in Duthé et al. (2025). Here, the geometric autoencoder is trained the pressure and 2D velocity field of Ik airfoil geometries with various inflow conditions. Once trained, the observation vector yo taken be noisy pressure observations at small number of nodes on the surface of the airfoil, emulating physical scenario when such pressure sensors may be attached to a wing/blade. The Bayesian inversion task is to reconstruct the pressure and velocity field around the airfoil, task far too ill-posed without highly informative prior. To compile the results in Table 3 we test GABI-ABC against the Direct Map with the same architecture type we randomize the number of observation locations to be drawn between and 50 observations with probability inversely proportional to the number of observation locations. For the supervised Direct Map approach the distribution of the number of observation must be known at train time, this is not the case for GABI whose training is
GABI-NUTS tested for 102 10 due to prediction runtime.
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Table 3: Comparison on Airfoil
Method
Field
MAE
% std
std
Train
Pred.
GABI-ABC
(p)
6.92 1072.00-22000 + 4.39 10-2
78.77%
97.28%
8.61hr
34.76s
Direct Map
(p)
5.35 1.35-10-=33.07-11
4.68hr
0.0040s
GABI-ABC
(Ua)
1.3.-10-14..22-0 -0-2 80.36%
97.28%
Direct Map
(va)
9.30-10-24..93
GABI-ABC
(vg)
3.94
2.92-00--2..0-00 10-2
75.87%
96.08%
Direct Map
(0g)
3.24
1.1 10-2
c9
co
(a) Pressure GT
(b) Pressure Mean
(c) Pressure Stddev.
(d) Pressure Error
Figure 3: Comparison of ground truth (GT), inferred mean, error, and standard deviation for pressure (p). The red dots correspond the observation location for the pressure. Full results including reconstruction of vertical and horizontal velocity fields are in Figure
Table 4: Comparison on Car Resonance Full Field Reconstruction and Source Localization
Method
Field
MAE
% std
2
Train
Pred.
GABI-ABC
(u)
2.32 10 + 7.21 10
75.48%
96.88%
2.29hr
18. 18.14s
Direct Map
(T)
6.03 6.03-10-1- -1 1.46 10
1.22hr
0.0032s
GABI-ABC
(
4.36
10-4
80.55%
97.49%
Direct Map
(F) 2.46 10-1 8.76 10-2
44
independent of the observation process. Figure 3 shows select number of results, For discussion on distrriiiitiiittoonn inference we refer the reader to Appendix B.4.2
4.3 CAR BODY ACOUSTIC VIBRATION AND SOURCE LOCALIZATION
Here, we study an acoustic resomance problem with a source localization task. We have a collection of car body geometries which undergo damped vibration due localized Gaussian bump forcing present in the first 1/5 of the vehicle (where the engine is). the inversion task is to reconstruct the full field acoustic vibration amplitude, u, as well as the full field foreing. f. on the surface of the vehicle from small number of sparse noisy measurements of the vibration amplitude. We do not observe the forcing field. The car geometries are taken from (Umetani Bickel (208)). We train model on 500 geometries, the number of observation locations is randomized in Section 4.2 The numerical results are in Table 4 and we show a reconstruction result in Figure
4.4 TERRAIN FLOW FIELD
To demonstrate the scalability of our approach, we apply it to the reconstruction of flow fields over complex terrain. Here, we use a large dataset (127GB) more than 5k RANS simulations, produced by Achermann et al. 2024b. From the original 64 grids, we extract 8-voxel-thick, terrain-conforming layers, resulting in 64 x 64 x 8 subdomains. Each subdomain is then converted into a graph, where voxels become nodes connected their adjacent neighbors. We scale both the training and the inference methods with multi-GPU pipelines. Using four RTX4090 GPUs, training takes 52hr, while inference takes 480s. Figure 5 shows the GABI-ABC pressure and velocity results.

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Published as a conference paper at ICLR 2026
u1
(a) n truth
(b) pred. mean
(c) pred. stddev
(d) u pred. error
(e) f truth
(0) pred. mean
(g) f pred. stddev
(h) pred. error
Figure 4: Ground truth, inferred mean, stddev., and emor for amplitude u, and forcing f. magenta are the observation locations.
(a) Pressure GT
(b) Pressure Mean
(c) Pressure Std
(d) Pressure Error
(e) ||l|| GT
(1)
|||||| Mean
||||| Std
(h)
||||| Error
Figure 5: Ground truth, inferred mean, error, and standard deviation for pressure and the magnitude of the velocity vector, ([/=]]) across two terrains. The red dots correspond the observation locations.
5 CONCLUSION
We propose a methodology for learning highly informative priors for solving Bayesian inverse problems across varying geometries. The scheme is realized through geometric autoencoder which learns informative latent representations allowing for sharing of information across geometries. Once trained this foundation model can be used with any observation process to perform Bayesian full field reconstruction over new geometries. A creative use of ABC sampling leads to efficient GPU implementation. The method achieves similar predictive accuracy to supervised direct map methods, all while outputting full Bayesian posterior over the fields of interest and being far more flexible due to the model being independent of the observation process. The method is tested on a wide range of physical setups and geometries.
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ACKNOWLEDGEMENTS
AV is supported through the EPSRC ROSEHIPS grant [EP/W00S816/1] MG is supported by Royal Academy of Engineering Research Chair and EPSRC grants [EP/X037770/1, EP/Y028805/1, EP/W00581 6/1,, EP/VO56522/1, EP/V056441/1, OP/T000414/1 and (P/R033710/1], GD and EC are supported by the BRIDGE Discovery Program of the Swiss National Science Foundation and Innosuisse (Grant No. 40B2-0 187087), as well as the French-Swiss project MISTERY funded by the French National Research Agency (ANR PRCI Grant No. 266157) and the Swiss National Science Foundation (Grant No. 2000211._212777
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A PROOFS
A.1 PROOF OF LEMMA 2.1
Proof. We remember
/ exp exp (x
7(y)
P. o P..
'u
Thus, assuming the relevant measurabilities,
Z(v)
exp ( (w; d(g
Ju
cxp (- (- (g((): diP,,
where the second equality is given by Theorem 3.6.1 in Bogachev Ruas 2007). Applying the same equality, VAA E F.
1
/ exp(-*(u::)) d(g
P (ul
Z(y)
Z(y) [ (x) dl)
Z(y) / 1^(g(>)) cxle)))xxpppppp
1
dP.
1
/ 1. (A) (2) exp (- dP:
7
7(v) /
exp exp(-
1(4)
-Pllo (A))
Po (A))))A)
where 1, a is the indicator function. Hence, P Puly = 94 Pely-
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A.2 PROOF OF THBOREM 2.2
Proof. Using Lemma 2.1
replacing 9 with D identifying 2, u, y with z, Ug Yo. obtain the n
desired posterior.
A.3 PROOF OF COROLLARY 2.3
Proof. This follows directly from Lemma 2.1 and Theorem 2.2 by identifying with (z, o.). ). P2 with
(I (..... y with (Dg @ Id.), changing the potential function to uu Joi Yo) - 2 lly H 2 and finally noting 1d. P..
IPo..
B ADDITIONAL NUMERICAL RESULTS AND DETAILS
The proposed methodology is architecture-agnotiii however for architecture to be appropriate for the task we require few particular qualities:
Geometry-aware architecture and data-structure. As the main premise of this work is to share information about physical fields which depend on the geometry of the problem to which they relate, it is essential that the data-structure reflect this geometric information, that the neural network architecture be effective processing it, and that both naturally accommodate inputs of varying size and topology.
Mapping to and from frromfixed-eee fixed-dimensional vectors. This framework relies on learning latent representations between solution to PDEs and the geometries which these defined. Hence, one needs to map variable-sizee sized geometric inputs to and from fixed-dimensional latent spaces.
Non-locality. All physical processes in this paper behave in non-local ways, the solution field one point in the domain affects the solution at all other points in the domain. Hence, when encoding solution fields for a given geometry, the encoder/decoder should reflect this non-locality.
We use three architectures, a Graph Convolutional Network (explained in B. Generalized Aggregation Network (explained in B.2) and transformer (explained and tested in B.3.3
B. GRAPH CONVOLUTIONAL NETWORK
We graph convolutional layers Kiif 2016) in the examples heat rectangles, RANS around airfoil, Helmholtz on car. To induce non- localiyy of mappings across channels, c, at each layer we GCN, : M,, x 2e M,, Re and the output is concatenated with a graph level channel-wise average which is expanded to populate each node of the graph. The edge attributes are the distance between nodes. The input node values always contain the node coordinates, and, for the encoder, also the data solution at the node.
In the encoder ES9 we map attributed graphs to latent vectors of fixed dimensions. The decoder maps a pairing between a latent variable and graph and populates the node attributes on the input graph. Hence the last layer in a channel-wist averaging across the graph and the channel dimension is taken to be the latent dimension. The decoder D takes as argument graph where the node attributes are only the coordinates, hence only contains geometric information, and we expand the latent sample z to decode to populate all nodes with a channel dimension equal to the latent dimension.
In GABI:
In Direct Map: For the direct map method we the same GCN layers as explained above. However now we must map observation vectors yn, and geometry M,, directly to the solution on the geometry, u, To do this, we make the input graph node attributes the coordinates of the nodes per the previous section, the observed nodal values yn. (where other values are 0), and a one-hot encoding indicates where this node is observed not.
GP Regression: We use graph Matérn Gaussian process (Mostowsky et al. 2024) with maximum marginal likelihood estimation of the hyperparameters.
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8161
10733
6.06
1072
600
0101
009
12
0088
ouea
(a) Ground Truth
(b) Predictive mean
(c) Predictive stddev
(d) Predictive Error
Figure 6: For the heat setup, we overlay in red the observation locations. (a) ground truth (b) decoded mode of predictive posterior. (c) empirical standard deviation of decoded posterior samples. (d) emro
between decoded mode and ground truth.
The rectangle, airfoil, and car examples are run on single RTX4090 GPU. The terrain example is run in multi-GPU manner.
B.2 GENERALIZED AGGREGATION NETWORK (GEN)
For the graph-based autoencoder applied the flow over complex terrain, replace standard GCN layers with Generalized Aggregation Network (GEN) layers (Li et al. 2020). This change allows us to add more complex geometrical information to the edge features (relative coordinates, similarly to (Plaff et al.) 20200). as GEN layers, unlike GCNs, can process multidimensional edge features. This message-passing formulation also uses roftmax-basee message aggregation scheme, which allows the model to dynamically weigh the importance of messages from neighboring nodes. Importantly, we maintain the same non-local pooling operation for each layer as for the GCN-based approach.
B.3 HEAT
In this section, we add numerical information and results for the steady-state heat problem. The equation used to generate the dataset
Au(x) = O, I Mn, u(x) = (x),
(9a)
I a....
(9b)
where hn described in Section 4.1 We use a finite element solver over an unstructured triangular mesh to solve the systems to be in the dataset.
B.3.1 IMPLEMENTATION DETAILS
Train: We use a 6-layer GCN as described in B.1 with 100 dimensional channel space and latent space. We use Ik training geometries and train the model for 20k iterations using the Adam optimizer (Adam al. 2014) with a learning rate of 10-3, We use a training batch size of 100.
Once trained, at inference time, GABI-ABC we decode 10k samples in 100 batches of 100 samples. We keep 100 samples being drawn from the posterior.
Pred:
B.3.2 ADDITIONAL RESULTS
In Figure 6 we show the results of inference with GABI-ABC on for the full field reconstructing from the small number of sparse noisy observations in red. In Figure 7 we show the two terms the loss for training the GABI autoencoder as well as a histogram of the encoded data batch overlaid with the ID standard normal distribution. We then test the performance of ABC sampling vs NUTS sampling as we vary the dimension of the latent space for the heat inversion task. The MAE, quality of UQ and inference times are shown in Figure g In Figure 9 we perform an ablation study on the size of the channel space of the graph autoencoder for fixed latent dimension of 100 for Bayesian inversion with both ABC and MCMC sampling. From this, we can assess the importance of network size for accuracy and efficiency.
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Histogrti
10
Lons
Standard Garedar
10
0
30000
100000
150006
200000
-10
-5
0
5
10
(a) Loss
(b) Latent histogram of encoded training batch
Figure 7: (a) Losse functions during training - total loss the sum these. (b) histogram
latent 1 across all dimensions after training.
384
MP
3TD 2a
(a) MAE
(b) UQ
(c) Inference time
Figure Sampling schemes on the same autoencoder model with the same channel space and different latent dimensions. We compare (a) mean absolute predictive error, (b) uncertainty quantification
under Gaussian assumption, (c) sampling time.
382
(a) MAE
(b) UQ
(c) Inference time
Figure 9: Sampling schemes the same autoencoder model with different channel space dimensions and same the latent dimension (fixed at 100). We compare (a) mean absolute predictive error, (b) uncertainty quantification under Gaussian assumption, (c) sampling time.
B.3.3 TRANSFORMER GABI
As our framework is architecture-agnostic we also test Transformer-basee variant. Transformers can function as neural operators (Cao) 2021), achieving non-locality through attention mechanisms. Unlike the graph-based approach, this variant treats the input as an unordered point set with coordi- nates, without exploiting the underlying graph structure. We implement 4-layer Transformers for the encoding and decoding networks with 64-dimensional embeddings and 8 attention heads per layer, resulting in a comparable number of trainable parameters: ~ 200k for GABI and ~ 100k for the direct map. As shown in Table 5 obtain similar results to the GCN-based approach (with the same training setup), but much slower training and prediction times, which can attributed to the dense self-attention operations of the Transformer layers.
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Table 5: Comparison of Graph Convolutional Network (GCN) and Transformer (T) architectures:
Heat Equation in Rectangular Domain
Method
Field
MAE
% 1 std
% 2 std
Train
Pred.
T: GABI-ABC
(u)
1.19 100-10-22000 + 8.65 10-3
82.97%
97.97%
18.89hr
10.17s
T: Direct Map
(u)
10-69-10-3
11.00hr
0.0063s
GCN: GABI-ABC
(u)
1.58-10-2+1
10-2
80.91%
95.59%
2.62hr
0.908s
GCN: Direct Map
1.25 1055-002+1.000 1.02
1.47hr
0.0029s
B.4 AIRFOIL.
B.4.1
IMPLEMENTATION DETAILS
Train: We use an 8-layer GCN as described in B.1 with a 100 dimensional channel space and latent space. We use 1k training geometries and train the model for 10k iterations using the Adam optimizer (Adam et al. 2014) with learning rate of 10-3 with an exponential decay learning rate with decay rate 0.999. We use a training batch size of 100.
Pred: Once trained, at inference time, in GABI-ABC we decode 50k samples in 100 batches of 500 samples. We keep 100 samples as being drawn from the posterior.
B.4.2 ADDITIONAL RESULTS
In Figure I0 we show four samples from the full field dataset. Figure I1 we show a complete example of results from the airfoil inversion task. We only measure the noisy pressure along the surface of the airfoil on the red scatter points and infer from this the full pressure field and 2D velocity field. In Figure 12 we show four random samples drawn from the generative prior model for given geometry, demonstrating the physical coherence and diversity of the prior.
We now test the behaviour of GABI-ABC inference 10 out-of-distribution airfoil geometries. In Figure 13 we plot a histogram of the density of the training dataset geometries and a selection of out of distribution airfoils. The x-axis shows the Mahalanobis distance (empirical covariance weighted distance of a point to the mean) between these shapes and the training dataset. The distance in geometry is computed using 9 summary statistics regarding airfoil thickness, camber, trailing-edgs thickness, etc. We report the MAE and percentage the ground truth between one and two standard deviations of the predictive posterior for pressure (reconstructed with GABI using 20 observations). These tests suggest that the GABI framework has some mild, but limited, robustness to prediction for outliers. Eventually, the miss-specification of the prior for such dfist--diittibution geometries may lead to unreliable inference.
B.5 CAR
For the car resonance and source localization problem we create the dataset by solving the damped Helmholtz equation (Liu al. 2024) a closed manifold
(AM. - k + irye) u(x)= = f(x), x Mp.
(10)
where = V-1, K 500 is a wave speed parameter, and 7 = 0.2 is a damping parameter.
B.5.1
IMPLEMENTATION DETAILS
Train: We use an 6-layer GCN as described in B.I with a 100 dimensional channel space and latent space. We use 500 training geometries and train the model for 10k iterations using the AdamW optimizer (Loshchilov & Hutter 2017) with an initial learning rate of 10-3 with and cosine decay learning minimum rate of 10 We use training batch size of 100.
Pred: Once trained, at inference time, in GABI-ABC we decode 50k samples in 100 batches 500 samples. We keep 100 samples being drawn from the posterior.
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32
208
(a) p. n
(b) Ve, 71
(c) Myy 1
3.0
C8 30
30
12 DO
020
12
(d) 2
(c) Par, 77 ===3 2
(f) Ug. = 2
40
3.
10.
150
150
(g) 3
(h)
72
(i) U/y
= 3
()) n
(k) Va. 7
Ug,
Figure 10: Four samples from the Airfoil dataset for pressure, horizontal and vertical velocity fields.
B.5.2 ADDITIONAL RESULTS
In Figure 14 we show some example geometries along with the forcing function and resonance field to reconstruct. In Figure I5 we show 3 random draws from the prior for given geometry to show the diversity of solutions and forcing fields.
B.6 TERRAIN
B.6.1
IMPLEMENTATION DETAILS
Train:
For both the encoding and decoding layers the autoencoder we use an encode encode-process- decode backbone (Pfaff et al. 2020 Duthé et al. 2025), where each processing block is composed
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0.0
0.0
0.0
120
284
0.00
003
0:3
-M4
005
0.44
-12
570
0.00
10
(a) Pressure GT
(b) U2
(c) V/y GT
022
0.0
022
022
000
800
0.22
20
0.30
00.0
0020
0.22
(d) Pressure Mean
(e) Ur Mean
(1) Uy Mean
027
0.0
022
300
080
18
2312
125
-0.0
228
025
(g) Pressure Std
(h) 0g Std
(i) Uy Std
677
26
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02b
100
0.30
6.36
-0.0
0000
6.12
4229
90
06
0.
(j) Pressure Error
(k) My Error
By
Error
Figure 11: Comparison of ground truth (GT), inferred mean, error, and standard deviation for pressure (p), horizontal velocity (vz), and vertical velocity (uy). Columns group physical variables; rows show different prediction outputs.
of 6-layer GEN-based GNN. Both edge and node features are projected to latent vectors with 64 dimensions. We train the model for around 300 epochs using the Adam optimizer with exponential learning rate decay (initial learning rate 10-3, -3, decay exponent of 0.99). This takes around 52 hours distributed training pipeline across four RTX4090 GPUs with overall batch size of 20 graphs.
We process the training samples from Joomemanaann al. (024), which larger (96 96 x 64) than the testing samples (64 x 64 x 64). Following their procedure, we apply random on-the-fi -ly crops and rotations during training. We then further reduce the samples by keeping only the 8 fluid cell
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19
19
139
83
-30
-150
-12
-18
-13
00
(a) p. 7
(b) Mhrs i
(c) By. =
185
175
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130
16
100
19
624
-125
-30
-20
-125
-13
(d) p. = 2
(c) U2, 8 = 2
(f) Ulys =
190
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13
10
300
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6.12
00
g p.
(h) Ma i 3
190
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60
in
60
p. 1- 4
By.
Figure 12: Airfoil field samples drawn from the geometry conditioned joint prior over pressure, Ug.
and by generated u, D l2
layers directly above the terrain, resulting final topography-conforming training geometry of 64 x 64 x 8.
Pred: inference time, we decode 50k samples in batches of 40 samples, distributed over GPUs, which takes around 480 seconds. We keep 100 samples being drawn from the posterior.
B.6.2 ADDITIONAL RESULTS
In Figure 16 show examples of flows different terrain geometries taken from the training set. In Figure 17 we show four random samples drawn from the generative prior model for a given terrain geometry, demonstrating the physical coherence and diversity of the prior. Due the
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Published as a conference paper at ICLR 2026
Training Geometries
OOD Testing Geometries
2std:
MAE: 0.167
MAE: 0.071 1std: 43% 25%
1std: 25%
2s0d: 53%
MAE: 0.362
MAE: 0.071 1std: 80%
MAE: 0.105
1sls: 36%
MAE: 0.229 1sto: 34%
2500: 61%
1stdi 35%
MAE: 0.193 59%
1snd:
2std: 80%
2stch 66%
Zeod: 10056
MAE: 0.065
1std: 78%
15 Mahalanobis Distance
25
30
Figure 13: Out of distribution airfoil shapes and corresponding GABI inversion MAE and UQ for pressure, based on 20 observed airfoil nodes.
179
130
(a)
(b) = 2
(c) = 3
2008
(n).
(e)
(0f,n=
Figure 14: Three example meshes in dataset
nature of the training data (inflows from all directions are possible), the generated samples are more diverse in nature than the previous numerical setups.
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Published as a conference paper at ICLR 2026
(a)
(b) u.
(c) u. i
(d) f
f.-22
f-
Figure 15: Given geometry, show 3 samples from the conditional prior generated as u, D
C DISCUSSION ON VAES
C.I ALTERNATIVE VARIATIONAL FORMULATIONS
In this work we propose to: first train an autoencoder to learn an informative prior the latent space, second sample from the posterior given observational data. At first glance, alternative approach would be to forgo this two step approach and attempt to directly learn a probabilistic geomerry-conditional autoencoder in the form of VAE
6" arg min ED, KL (02x. llp.)l....
(11)
0,P
KL 9 12ly Paly log Py., (yn) + B [- log Py,,y (yr)) + KL (oh.y..l12
(12)
Upon close inspection the loss for such model, is apparent that information from the full solution field u, is not incorporated in the learning, hence one cannot hope to learn the correct relationship between yn and u,
One could alternatively train model of the form
0* 3
arg min ED, KL ( IPp.....))
(13)
0.0
However, as u, are each associated with different geometries Mn we cannot put common prior distribution over these in a statistically interpretable manner. Furthermore, such a model would not learn the correct relationship between yn and u, as observational data ingested in an unsupervised manner, regression loss encourages to be close to u, in any way.
There exist other models such as UQ-VAEs VAEs (Goh et al.) 2022) where the decoder is replaced by the solution operator for the assumed PDE responsible for generating the data. This of course requires us to know the governing equations of the problem, the boundary conditions etc and has the same challenges and limitations as other physics-informed methodologies. Furthermore, this framework is not made to handle variable geometries.
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Published as a conference paper at ICLR 2026
(a) p. n-
(b) Up. 72
(c) Ug. nt
(d) U2. n
(e)
(f) Uz.
(g) Ug. n
(h) V2.
W p.
ii Vo. 3
(k) Ug.
(1 V.. 3
(m) n = 4
(n) n
(0) Ug.
(p) Us. 4
Figure 16: Four examples of flows (p. Us. Uy U2) over complex terrain taken from the training
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Published as a conference paper at ICLR 2026
p.)
(b) i = 1
(c) Uy. i= 1
(d) Us i = 1
(c) p. 1
(f) Ur.
(g) By. i 2
(h) U2a 2
(1 p. 3
U2,
Myy 3
U2.
(m) p. = 4
(n) C.. i = 4
(0) Ty. i 4
(p) i 4
Figure 17: Flow field samples for a given test terrain drawn from the geometry conditioned joint prior over p. Urs By. and Uz, generated as u, # lz
25


Reviews:
[{"review_id": "8QSmTmVlfL", "reviewer": "Reviewer_WHwu", "summary": "The paper introduces GABI (Geometric Autoencoder Priors for Bayesian Inversion), a framework for solving inverse problems, such as reconstructing temperature, pressure, or flow fields from limited noisy measurements, when system geometries vary. Instead of assuming known physics or PDEs, GABI learns a geometry-aware prior via an autoencoder that compresses physical fields across diverse geometries into a shared latent space. During inference, it performs Bayesian inversion in this latent space using Approximate Bayesian Computation (ABC) sampling, which scales efficiently on GPUs and outputs well-calibrated uncertainty. The approach enables a “train once, use anywhere” model that generalizes to unseen geometries and observation setups. Experiments across heat transfer, airfoil flow, car acoustics, and large-scale terrain flow show GABI matches supervised baselines in accuracy while offering full Bayesian uncertainty and far greater flexibility.", "questions": "How robust is the learned latent geometry field representation when the target system exhibits physical behaviors outside the training manifold, such as topological changes, discontinuous boundaries, or new forcing regimes? And can GABI’s prior adapt online without retraining to preserve its Bayesian consistency?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "I think what makes this paper stand out isn’t just that it “does Bayesian inversion with deep learning”, but how it redefines the workflow. Most prior works either fix geometry and bake the PDE into the model, or treat inference as a supervised regression from observations to fields. Both approaches that crumble when geometry or observation setup changes. GABI breaks this dependency entirely: it learns a geometry-conditioned latent prior that can be reused across arbitrary domains, making it like an attempt for a foundation model for physical inversion. The theoretical guarantee that latent-space inference is equivalent to inference in the original field space is elegant and nontrivial, anchoring the method in solid Bayesian footing, rather than just heuristics.\n\nAnother secondary (yet distinctive) strength is the combination of MMD-based autoencoding and ABC sampling, which replaces the usual VAE–MCMC pairing with something GPU-parallelizable and architecture-agnostic, making large-scale problems (like the multi-GPU terrain flow case) actually tractable. \n\nFinally, GABI’s decoupling of learning from the observation process is a rare and important contribution (it may have been proposed before but I am not 100% sure, other reviewers could correct me if I am wrong), most real engineering problems involve messy, variable sensors and noise models, and this paper’s “learn once, observe later” philosophy directly addresses that reality instead of hand-waving around it.", "weaknesses": "The paper seems to lean heavily on the assumption that a single latent space can meaningfully represent diverse geometries, but offers little analysis of when this assumption holds or how latent alignment behaves as geometry complexity scales, especially in high-frequency, discontinuous regimes (e.g., shocks or turbulence).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1FcDRDZ2pP", "reviewer": "Reviewer_WWiD", "summary": "This is an interesting work applying Bayesian methods (autoencoders) to Geometric (prediction) problems. This will be applied when a new shape is designed, and people want to know how the field (temperature, pressure) will change. The authors use a simple solution that just encode the shapes (M) into a latent space during learning. When given a new shape, suppose we have enough trained shapes - the model will sample in the latent space N(0, 1), and produce related vector fields from the decoder, and match with the available observations. From four experimental study, the performance is verified.", "questions": "1. In 1.1 setup, would the authors define U? In Lemma 2.1 or 1.3, would the authors use a line or two to describe what z normally is (e.g. the latent space)? \n\n2. In Figure 2, the reviewer would like to ask (i) why is this figure so small.. and (ii) how should we interpret (b)? \n\n3. The reviewer is very curious if the computation can be accelerated without incorporating the Bayesian framework. It seems that the Bayesian framework is important, but might not be that essential (if one wants to have the fastest algorithm). \n\n4. How would GABI perform for OOD shapes? The reviewer believes that OOD it will perform poorly as the decoder can have random behavior during extrapolation. A comment here would be great (don't need to change paper).\n\n5. Would the authors add some words about the implicit assumption in the main paper? As the latent space z might not actually exist. Or - the latent space z might be able to represent U/M under a bounded area/domain. With better/larger collected data, this problem can be resolved. \n\nThe reviewer is likely to improve the rating if some of the comments above being resolved.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "I like this paper. It proposed a very valid method using autoencoders, and potentially solves a pretty difficult and highly practical problem. \n\n1. The presentation is very nice. The written style, mathematical notation, experimental figures are all well presented. \n\n2. The idea is clean, and nicely described. The algorithm is simple and easy to understand. \n\n3. The experimental study nicely demonstrates the advantage of this method.", "weaknesses": "The theory part looks strong, but may have very limited theoretical impact. Although everything is well documented in theory, but it could create problems for readers. And an important critique from the reviewer is that - the theory seems to be rooted on two things (i) such latent space exists and (ii) the decoder (from this latent space) can be expressive enough, and it is able to nicely extrapolate data. These are all likely to be true, yet unfortunately, these assumptions are potentially hidden in the problem statement. \n\nPlease see questions below as well.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "t3YfW5ods4", "reviewer": "Reviewer_RTKB", "summary": "This paper combine autoencoder (AE) and approximate Bayesian computation (ABC) for inverse problems. AE is trained on geometry of the physics problems to obtain common features in the latent space, which is independent of the governing equations (PDE). The observation process is only used in the sampling stage with ABC. The proposed method, hence named Geometric Autoencoders for Bayesian Inversion (GABI), enables efficient Bayesian inversion in a `learn first, observe later` fashion.", "questions": "1. How large the geometric AE is required to ensure sufficient precision in ABC? Is there any trade-off between the size of AE and accepted sample size $N_a$? Can you do some ablation study on it?\n\n2. Since the AE is trained on the geometry of different but similar physics problems, I wonder how GABI relates to or compares with transfer learning methods?\n\nMinor things:\n\nLine 062: remove semicolon.\nLine 098: I assume you meant for $H_o: R^{d_u} \\to R^{d_o}$?\nLine 119, 143: period is missing after the reference.\nLine 422: `Figure` 4.\nAfter Equation (6), maybe emphasize that the likelihood is not computed but ABC is used.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The idea is straightforward, clearly presented, and efficiently implemented (via multi-GPUs). The demonstration on high-dimensional real engineering problems (car body acoustic vibration) is definitely a plus.", "weaknesses": "It is not clear the trade-off between the size of geometric AE and the precision of the downstream ABC. And it is not clear how it compares with transfer learning method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "NcakGUdjSG", "reviewer": "Reviewer_TrCV", "summary": "The paper proposed an autoencoder-based scheme of constructing priors for inverse Bayesian problems in PDE. These priors are build from data of similar tasks with the same underlying physics but different meshes.", "questions": "l. 96: which space is $u_o$ from?\nl. 97: What kind of Operator is $H_o$? Linear? Bilinear? \nl. 97: $R^o$ is the output of $H_o$. Why/how does it select nodes?\nl. 198: How do you compute the argmin? (SGD does not compute argmin!)\nl. 207: what is $H_o$? What is $D_o$?\nl. 211-215: what are the various $p$'s introduced here?\nl. 255ff: I do not even know which part of the methodology is trained using neural networks. Encoder? Decoder? $H_o$? All of them?", "limitations": "", "rating": 2, "confidence": 2, "soundness": 1, "presentation": 1, "contribution": 3, "strengths": "The strength in this paper lie in strong experimental evaluation. The results of the experiments seem good and improve upon the state of the art by up to an order of magnitude. The Experimental range is quite wide, starting from the simple heat equation to more complex, non-linear systems of PDEs in varying geometries. This is the main reason for a good contribution score.", "weaknesses": "The main weakness lies int he presentation of the paper. I had quite a hard time of following it and even at this stage I do not get the details of the approach.\n - Some problems lie in formalizing the notation. See the questions below. I stress that these questions are just for the main paper, the appendix (which I just flew over) was equally problematic. I want to clarify that for many of my questions bwloe it is alluded what the notation means, but not made formal.\n - The paper solves a relevant problem. However, it took me quite some time to understand (and then only even vaguely) which problem is being solved. My main problem was that it took some time to understand what was given and what was not given.\n - Experiments compare to machine learning methods only, and not to methods from the inverse problems literature. (I know that it is hard to do experiments against all potential competitors, so I am not dwelling on this point for so long, as some comparisons are being made.)\n - I did not find any description of computational time in this paper. For such computational approaches, I think this is a critical point.\n - l. 256ff \"The proposed methodology is agnostic to any particular architecture, however we need the following characteristics: Geometry aware architecture and data-structure; Mapping to and from fixed-dimensional vectors; Non-locality. We expand on these points in B.\" This makes the paper (without appendix) far from being standalone.\n - Lemma 2.1 is relatively simple. However, it is stated very technically, without giving an intuition. \nThese weaknesses explain my soundness and presentation scores.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)