The Problem With Judging Inverse Solvers by Their Best Guess
In scientific computing, inverse problems are everywhere. You have sparse sensor readings from deep underground and need to figure out what the rock looks like. You have blurry images through clouds and need to infer the material inside. You observe CO2 behavior at a few monitoring wells and need to characterize the entire subsurface. The forward physics is often expensive, nonlinear, and many-to-one: many different physical configurations can produce the same observations.
This non-uniqueness is not a bug. It is the fundamental nature of the problem. A meaningful answer is not a single reconstruction but a posterior distribution, the full set of physically plausible configurations consistent with the data. When generative models, particularly diffusion-based samplers, are applied to these problems, they are theoretically capable of producing samples from this posterior. But most evaluations still ask a simpler question: how close is the single best reconstruction to the ground truth?
That question is insufficient. A solver can achieve excellent pointwise accuracy while collapsing multiple modes into one, producing overconfident uncertainty estimates, or averaging incompatible solutions into an unphysical blur. None of these failure modes show up in standard reconstruction metrics.
PosteriorBench, introduced by researchers from Caltech, National Taiwan University, Lawrence Berkeley National Lab, Stanford, Imperial College London, and EarthFlow AI, addresses this gap. Instead of evaluating a solver's best sample, it evaluates whether the solver's generated ensemble matches the actual posterior distribution.
Four Physics Tasks, Each With a Reference Distribution
The benchmark covers four inverse problems that differ in their priors, observation patterns, forward models, and sources of ambiguity.
Darcy flow inversion recovers a binary permeability field from sparse pressure measurements. The prior is a thresholded Gaussian random field producing discontinuous high- and low-conductivity phases. Multiple connected pathways can produce similar pressure readings at observed locations, making the posterior inherently multi-modal. The ambiguity is structural: which spatial arrangement of conductive channels is consistent with the data?
Poisson source recovery infers a continuous source term from sparse observations of the resulting potential field. The prior uses Gaussian random fields with varying correlation length and smoothness, sampled independently per case. This tests whether solvers generalize across a broad family of prior spectra rather than overfitting to a single fixed prior structure.
Carbon capture and storage (CCS) is a high-impact subsurface monitoring setting. Given sparse well-column observations of CO2 saturation dynamics, the task is to characterize the underlying permeability field. Geostatistical priors generate multimodal permeability realizations, and a reservoir simulator produces the corresponding saturation fields. Many geomodels can match the same well measurements, leaving substantial spatial uncertainty away from the wells. Reducing the number of monitoring wells directly lowers field intervention and monitoring cost, making accurate uncertainty quantification practically valuable.
Light transport material inference (LTMI) tackles a two-layer participating medium problem motivated by atmospheric imaging, optical tomography, and nondestructive inspection. Low-resolution optical measurements must be translated into spatially varying extinction coefficients. Different arrangements of scattering material can produce similar aggregate measurements under limited observation, and the high-dimensional unknown field combined with severely restricted measurements makes this a particularly challenging inference setting.
Building Reference Posteriors the Hard Way
The core design principle is that each benchmark case must come with a known reference posterior against which solvers can be compared. This is computationally expensive but necessary. You cannot evaluate distributional matching without knowing the target distribution.
For most tasks, the reference posterior is constructed through rejection sampling. A large candidate pool is drawn from the prior, each candidate is simulated through the forward model, and candidates are weighted by their mismatch to the observed data under a Gaussian noise assumption. The soft likelihood weights take the form of an exponential penalty on the squared observation error, scaled by the noise variance. For efficiency, only candidates within three standard deviations of the observation are retained, producing a weighted ensemble of 100 reference samples. These reference posteriors are validated for stability and observation consistency before use.
The computational cost of this procedure is precisely why existing benchmarks avoid it. Rejection sampling for the CCS task, for instance, requires running a reservoir simulator thousands of times. PosteriorBench makes this investment and provides the resulting distributions as part of the benchmark, removing the burden from solver developers.
Five Metrics That Catch Different Failure Modes
The evaluation suite uses five metrics designed to assess different aspects of posterior quality.
Posterior-mean error measures whether the solver's ensemble center agrees with the reference ensemble center. This is a relative L2 error, not a pointwise comparison against a single ground truth, so it captures systematic bias in the expected field.
Posterior-standard-deviation error measures whether the solver captures the correct magnitude of pointwise uncertainty. A solver that underestimates variance will have a small mean error but a large std error. The paper emphasizes that these two metrics should always be read jointly: a solver can match the mean while misrepresenting the spread.
Maximum mean discrepancy (MMD) detects discrepancies in higher-order spatial statistics using a multi-scale RBF kernel with bandwidths of 0.2, 0.5, 1.0, 2.0, and 5.0 times the median squared distance. This makes the statistic sensitive to both coarse structural shifts and finer spatial patterns that moment-based metrics miss.
Sliced Wasserstein distance (SWD) evaluates geometric proximity by projecting samples onto smooth random directions drawn from a Gaussian random field distribution. The GRF projection favors spatially coherent test functions over random pixel-wise projections, aligning the metric with physical field discrepancies. The fields are z-score normalized before projection to handle different physical units and dynamic ranges across tasks.
Radially averaged power spectrum error (RAPS) evaluates spectral consistency, whether the solver preserves the energy distribution across spatial scales. This catches a common failure mode: over-smoothed samples that match low-frequency structure but lose the fine-scale texture present in the reference posterior. The geometric mean of per-bin relative errors avoids overemphasizing high-frequency bins with low power.
What the Experiments Revealed
Eight solvers were evaluated: function-space diffusion methods (FunDPS, Fun-DDPS, DDIS, FunDiff), a PDE-constrained diffusion method (DiffusionPDE), an ensemble data-assimilation baseline (ES-MDA), a sampling method with hard replacement (ECI-sampling), and a direct neural uncertainty baseline (FNO with MC Dropout).
The most striking finding is that pointwise accuracy and posterior quality often diverge. On the LTMI task, FNO with MC Dropout achieves lower posterior-mean error than FunDPS, but its posterior-standard-deviation error, MMD, and SWD are all substantially higher. Visual inspection shows why: MC Dropout samples are visibly over-smoothed relative to the reference, capturing the mean field while missing the spatial texture and spread. The V-shaped relationship between pointwise error and distributional metrics across methods and tasks confirms this is not a fluke. Aggressively fitting a single reference field can degrade posterior structure.
Function-space diffusion samplers, particularly FunDPS, Fun-DDPS, and DDIS, are generally strong performers across tasks. Their function-space score priors, built on spectral operator backbones, are better aligned with continuous physical fields than grid-based alternatives. But performance is task-dependent. On LTMI, ES-MDA (a classical ensemble assimilation method) is competitive, and the best function-space diffusion sampler varies across datasets. The comparison with DiffusionPDE highlights the value of the function-space backbone, and the comparison with ECI highlights the role of soft likelihood guidance: ECI enforces observed entries through hard replacement, while guided diffusion samplers condition the sampling process while staying on the prior manifold.
Noise, Guidance, and the Calibration Tradeoff
The paper's noise-level ablation reveals a fundamental tension. For a Gaussian observation model, guidance weights should theoretically scale with the inverse observation-noise variance. In practice, learned samplers do not resolve calibration through this scaling alone.
Stronger guidance improves observation consistency, pulling samples closer to what the sensors measured, but often underestimates posterior variance. The sampler becomes overconfident, producing tight clusters around a biased mean. Weaker guidance preserves diversity at the cost of a biased or weakly conditioned posterior mean. The ensemble spreads out but drifts away from the data.
This mean-variance tradeoff is not a hyperparameter tuning problem with a clean solution. It motivates conditioning mechanisms that can calibrate posterior mean and uncertainty jointly, rather than relying on a single guidance weight to do both jobs. The current generation of solvers does not have such mechanisms, and this is identified as a key direction for future work.
Resolution Robustness and Practical Implications
The resolution ablation shows that neural operators improve robustness to observation resolution. Function-space diffusion methods maintain posterior quality as resolution decreases, while grid-based methods degrade more quickly. This matters for practical applications where sensor resolution is limited by hardware or cost constraints.
For practitioners working on subsurface monitoring or optical imaging, the benchmark results suggest a specific workflow. Use function-space diffusion samplers as the default choice, particularly when observations are sparse or low-resolution. Tune guidance weights with awareness of the mean-variance tradeoff, not just observation consistency. And evaluate using the full metric suite, not just mean error, to avoid the trap of a solver that looks good on reconstruction but fails on distribution.
The code is available at https://github.com/neuraloperator/PosteriorBench, providing the reference posteriors, evaluation pipeline, and baseline configurations needed to evaluate new solvers against the same targets.
What This Means for Generative Modeling in Science
PosteriorBench reframes the evaluation question from "how good is the reconstruction?" to "does the solver actually sample from the posterior it claims to?" This is not an academic distinction. In scientific settings, posterior uncertainty drives decision-making: whether a CO2 storage site is secure, whether a material inspection reveals defects, whether a subsurface model supports a particular engineering intervention. A solver that produces accurate point estimates but wrong uncertainty is worse than useless for these decisions, because it provides false confidence.
The benchmark's emphasis on reference posteriors constructed from first principles also sets a higher standard for the field. Previous benchmarks either evaluated point estimates or used heuristic quality measures. PosteriorBench demands that every claim about posterior quality be backed by comparison against a known target, computed at significant computational expense. This makes the evaluation reproducible and comparable across methods.
The persistent distribution-matching gaps the authors identify across current solvers suggest that the generative modeling community has been optimizing the wrong objective. Improving pointwise reconstruction accuracy while ignoring distributional fidelity is a local optimum. The path forward likely involves architectures that jointly model the posterior mean and its spatially varying uncertainty, rather than treating them as separate problems to be solved after sampling.