Bayesian inference via Markov Chain Monte Carlo (MCMC) provides effective parameter estimation, but its real-time application in complex physical systems is hindered by heavy computational bottlenecks and extreme sensitivity to statistical noise. We address this by proposing a neural-network-based probabilistic surrogate framework for rapid and robust MCMC inference. Using fusion plasma Thomson scattering diagnostics as a challenging, noise-dominated testbed, our approach employs a dual-head architecture to simultaneously estimate the expected physical emission spectrum and the channel-wise intrinsic measurement noise variance. By optimizing a Gaussian Negative Log-Likelihood (GNLL) objective, the learned aleatoric uncertainty dynamically buffers the sampler against pathological shot noise. Evaluations demonstrate that this surrogate framework achieves > 1500x acceleration over exact physical forward models, while simultaneously reducing inference error (RMSE) by >20% compared to standard homoscedastic neural baselines, offering a highly promising paradigm for real-time physical analysis.
Figures & tables
Figure 1. Comparative overview of MCMC inference in extreme-noise plasma diagnostics. Top: The forward model F(θ) maps latent VDFs to noisy spectra. Markers: green star (ground truth), yellow circle (prediction), and blue ellipse (posterior uncertainty). A: Physical MCMC yields theoretically exact posteriors but is computationally prohibitive and highly noise-sensitive (slender ellipse). B: Standard NN Surrogate (homoscedastic MSE loss) is fast but yields structurally biased and overconfident posteriors (circle far from star). C: Proposed Heteroscedastic Surrogate (GNLL loss) predicts data-dependent log-variance. This learned uncertainty acts as a topological smoother, penalizing overconfident proposals to enable fast and stable MCMC inference (ellipse’s appropriate shape, nearly aligned circle and star).
Figure 2. Workflow of the proposed heteroscedastic neural surrogate and its MCMC integration. (1) Neural Surrogate: A backbone processes abstract physical parameters θ (latent VDF) via a dual-head network, predicting expected spectrum μGNLL(θ) and channel-wise log-variance sGNLL(θ) . (2) GNLL Optimization: The network is trained via Gaussian Negative Log-Likelihood (GNLL) loss, where learned aleatoric uncertainty attenuates residual penalties in high-noise regions. (3) MCMC Inference: The frozen surrogate evaluates the heteroscedastic log-likelihood. The learned variance regularizes the Metropolis-Hastings acceptance ratio, yielding a robust, physically consistent posterior.
Figure 3. Qualitative comparison of 2D Posterior Probability Density Functions (a: best seeds with minimal absolute Te error, marked † in Table 1 ) and parameter trace plots across 20 independent seeds (b, c). Method A: yields precise ‘best-case’ estimates, though 95% credible intervals remain widely dispersed due to extreme noise sensitivity. Method B: collapses into a biased and overconfident geometry (almost vanishing error bars) by overfitting localized shot noise. Method C: Our proposed GNLL Surrogate acts as a topological smoother, bypassing noise-induced local minima. Its 95% credible intervals consistently encompass ground-truth values, ensuring the estimated uncertainty aligns with actual estimation errors under extreme noise.
Test ID
Method A (Physical MCMC)
Method B (Standard MSE)
Method C (Proposed GNLL)
Te (eV)
ne(×1018 m−3)
Te (eV)
ne(×1018 m−3)
Te (eV)
ne(×1018 m−3)
Seed 01
6000.18†
2.008†
5975.22
2.004
5948.87
1.998
Seed 02
5895.83
1.979
6002.32†
1.994†
6017.00
1.997
Seed 03
5984.37
2.007
6051.74
2.019
6139.09
2.032
Seed 04
5894.42
1.987
6002.89
2.001
6060.64
2.013
Seed 05
6015.62
2.003
6022.20
2.002
6020.44
2.002
Table 1. Comprehensive MCMC inference results across 20 independent extreme-noise blind tests (Target Truth: Te=6000 eV,ne=2.000×1018 m−3 ). The summary metrics—Ensemble Std, RMSE to Truth, and Inference Time per posterior PDF—quantitatively demonstrate that the proposed GNLL surrogate (Method C) achieves ∼1500× acceleration over the physical MCMC (Method A) while surpassing the standard MSE (Method B) in both accuracy (RMSE) and stability (Std). Values marked with a dagger ( † ) denote seeds visualized in Figure 3 (a), chosen for minimum absolute Te error.
Variant
Variance Modeling
MCMC Likelihood
Std ( Te )
Std ( ne )
RMSE ( Te )
RMSE ( ne )
Standard MSE
None
Uniform
103.43
0.0179
105.19
0.0179
Global GNLL
Global Scalar
Uniform
90.51
0.0145
88.29
0.0146
Decoupled GNLL
Channel Vector
Uniform
87.48
0.0143
85.84
0.0144
Proposed GNLL
Channel Vector
Data-Dependent
78.88
0.0141
83.76
0.0144
Table 2. Ablation study decoupling the architectural and inferential components of our framework. We evaluate the necessity of channel-wise variance modeling and its explicit integration into the MCMC likelihood evaluation. Metrics isolate the inference precision (unbiased Std and RMSE), with units being eV for Te and ×1018 m−3 for ne .
Bayesian inverse design provides a principled framework for inferring aerodynamic geometries from sparse flow observations while quantifying uncertainty. However, its practical use in computational fluid dynamics (CFD) is severely limited by the cost of repeated high-fidelity simulations required for gradient-based Markov chain Monte Carlo (MCMC) sampling. While surrogate models are commonly proposed to reduce this cost, their effect on posterior geometry and uncertainty, especially for shock-dominated flows, remains poorly understood. In this work, we demonstrate that neural operator surrogates can be embedded directly within the MCMC inference loop while preserving posterior structure. Using a fully Bayesian inverse formulation of quasi-one-dimensional nozzle flow, we demonstrate that geometry parameterization plays a decisive role in identifiability and posterior conditioning, with cubic B-splines yielding stable and physically meaningful uncertainty estimates. Building on this formulation, a Deep Operator Network trained on CFD-generated data is substituted for the CFD solver within a No-U-Turn Sampler, while keeping the likelihood model, priors, and sampling configuration unchanged. Across sparse to fully observed regimes, surrogate-based inference reproduces the posterior geometry and uncertainty trends of the CFD reference. As a result of surrogate integration, total inference time is reduced to under one second, corresponding to a speedup exceeding three orders of magnitude. In addition, a direct inverse neural operator is examined as a deterministic alternative for inverse design, enabling single-shot geometry reconstruction without posterior sampling. These results demonstrate that neural operator-accelerated Bayesian inference enables practical, uncertainty-aware inverse design workflows for aerodynamic applications.
Bipin Tiwari, Omer San
Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville,2026 TN 37996, USA
Accurate condition monitoring of industrial equipment requires inferring latent degradation parameters from indirect sensor measurements under uncertainty. While traditional Bayesian methods like Markov Chain Monte Carlo (MCMC) provide rigorous uncertainty quantification, their heavy computational bottlenecks render them impractical for real-time process control. To overcome this limitation, we propose an AI-driven framework utilizing Simulation-Based Inference (SBI) powered by amortized neural posterior estimation to diagnose complex failure modes in heat exchangers. By training neural density estimators on a simulated dataset, our approach learns a direct, likelihood-free mapping from thermal-fluid observations to the full posterior distribution of degradation parameters. We benchmark this framework against an MCMC baseline across various synthetic fouling and leakage scenarios, including challenging low-probability, sparse-event failures. The results show that SBI achieves comparable diagnostic accuracy and reliable uncertainty quantification, while accelerating inference time by a factor of82× compared to traditional sampling. The amortized nature of the neural network enables near-instantaneous inference, establishing SBI as a highly scalable, real-time alternative for probabilistic fault diagnosis and digital twin realization in complex engineering systems.
Peter Collett, Alexander Johannes Stasik, Simone Casolo +1
Cognite AS, Oslo, Norway · Department of Data Science, Norwegian University of Life Sciences, Ås, Norway · Department for Mathematics and Cybernetics, SINTEF Digital, Oslo, Norway
Three-dimensional probabilistic inversion of time-domain airborne electromagnetic (AEM) data is limited by the cost of the forward solve. Even though one simulation takes only tens of seconds, a Bayesian inversion of a survey of millions of soundings requires of order 1010 forward evaluations. To address this, we develop a continually learning neural-operator surrogate of the three-dimensional AEM forward operator that replaces the solver inside the Bayesian inversion. We start from the point of view that regardless of what geological prior is specified, Maxwell's laws remain invariant. Secondly, we avoid the limitation of learning on a single prior by continual learning on consecutive priors, which means our surrogate becomes richer as it is applied in future case studies, either by the authors, or by the scientific community. We use a validity check built on ensemble disagreement to divert cases with measurements outside the training range to the solver. Driven by the surrogate, the identical Markov chain Monte Carlo sampler reproduces the full-solver posterior, and its credible intervals cover the truth within 2.6 percentage points. Applied to the 2013 Capricorn TEMPEST survey in Western Australia, the surrogate inverts over two million soundings in seconds, a computation infeasible for the solver. Testing the geological prior against the entire survey costs minutes. The framework delivers uncertainty-quantified conductivity imaging at survey scale, which we believe is essential to perform near real-time mineral-systems targeting with geophysics.
Jaehong Chung, Andrew Lockwood, Jef Caers
Mineral-X, Department of Earth and Planetary Sciences, Stanford University, USA · Xcalibur Smart Mapping, Perth, Australia