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Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons

Apostolos F Psaros, Xuhui Meng, Zongren Zou, Ling Guo, George Em Karniadakis

arXiv:2201.07766v1cs.LG

TL;DR

Scientific machine learning addresses ill-posed physical-system problems, but neural-network uncertainty quantification must handle noisy data, limited data, and other uncertainty sources. This paper develops and compares a comprehensive UQ framework across PDE, stochastic, and operator-learning problems, finding that several cheaper methods can match more expensive ones while calibration and uncertainty coverage remain important concerns.

  • Problem

    Accurate prediction of dynamic physical systems remains challenging for ill-posed problems, noisy or gappy data, and high-dimensional computational settings.

  • Method

    The paper combines uncertainty models, existing and new inference methods, evaluation metrics, post-training calibration, learned functional priors, and comparative studies across physical and stochastic problems.

  • Results

    Across experiments, cheaper methods such as MFVI and LA achieved performance comparable to more expensive HMC and deep ensembles, while GAN-FP and related methods improved selected PDE and operator-learning results.

  • Takeaways & Limitations

    UQ method assessment should use multiple evaluation metrics, and small left-out datasets can substantially improve calibration through post-training calibration.

  • Takeaways & Limitations

    In the mixed SPDE experiments, ensemble-based epistemic uncertainty did not always cover errors in predicted first- and second-order statistics and could produce over-confident predictions.

Abstract

from arXiv · show

Neural networks (NNs) are currently changing the computational paradigm on how to combine data with mathematical laws in physics and engineering in a profound way, tackling challenging inverse and ill-posed problems not solvable with traditional methods. However, quantifying errors and uncertainties in NN-based inference is more complicated than in traditional methods. This is because in addition to aleatoric uncertainty associated with noisy data, there is also uncertainty due to limited data, but also due to NN hyperparameters, overparametrization, optimization and sampling errors as well as model misspecification. Although there are some recent works on uncertainty quantification (UQ) in NNs, there is no systematic investigation of suitable methods towards quantifying the total uncertainty effectively and efficiently even for function approximation, and there is even less work on solving partial differential equations and learning operator mappings between infinite-dimensional function spaces using NNs. In this work, we present a comprehensive framework that includes uncertainty modeling, new and existing solution methods, as well as evaluation metrics and post-hoc improvement approaches. To demonstrate the applicability and reliability of our framework, we present an extensive comparative study in which various methods are tested on prototype problems, including problems with mixed input-output data, and stochastic problems in high dimensions. In the Appendix, we include a comprehensive description of all the UQ methods employed, which we will make available as open-source library of all codes included in this framework.

1. Introduction

The paper addresses uncertainty quantification for neural scientific machine learning across neural PDEs, stochastic PDEs, and neural operators. It develops a unified framework combining uncertainty modeling, UQ methods, evaluation metrics, post-training improvement, and comparative studies.

  • Motivation and scope: SciML combines observational data with physical models through emulation, physics-informed architectures, or physical constraints in NN optimization.These approaches target ill-posed problems, noisy or gappy data, and computationally costly high-dimensional systems.
  • Framework and methods: The framework integrates posterior inference, prior learning, data-noise modeling, uncertainty evaluation metrics, and post-training calibration across PINNs, neural operators, and SPDE solvers.It is designed to combine physics with new and historical data affected by different noise types.
  • Problem formulation: The paper considers four scenarios: forward deterministic PDEs, mixed deterministic PDEs, mixed stochastic PDEs, and operator learning.Mixed problems seek both solutions and partially unknown parameters, while operator learning separates pre-training from inference.
  • Novel contributions: Functional priors use historical data to address function approximation with heteroscedastic noise, while a Gaussian-process–GAN combination provides a new method for deterministic forward PDEs.The paper also treats mixed PDEs with heteroscedastic noise in source, parameter, and solution data.
  • Novel contributions: The study solves mixed SPDEs with noisy stochastic realizations and proposes a neural-network architecture using polynomial chaos for uncertainty quantification.It additionally addresses noisy and incomplete inference data after pre-training a neural operator on clean data.
  • Novel contributions: The paper proposes methods to detect out-of-distribution data in neural operators, motivated by risk-related applications.These methods complement the broader goal of making UQ routine in scientific machine learning.

2. Neural PDEs and neural operators

The section presents PINNs for solving neural PDEs and DeepONets for learning operator mappings, with both architectures combining neural approximators with data or physical structure. PINNs use shared parameters and differential operators, whereas DeepONets combine branch and trunk networks through an inner product.

  • Neural PDEs: PINNs construct neural approximators for PDE solutions and use differential operators to produce modeled source and boundary terms.The solution approximator is differentiated automatically, and the resulting terms are fitted to the dataset through optimization.
  • Neural PDEs: Shared parameters connect the PINN approximators uθ, fθ, and bθ, which are trained simultaneously to fit the complete dataset.The relationships are fθ = Fλ[uθ] and bθ = Bλ[uθ].
  • Neural operators: DeepONet learns mappings from input functions λ(x; ξ) to solutions u(x; ξ) using separate branch and trunk subnetworks.The branch net processes λ, while the trunk net processes the location x where the solution is evaluated.
  • Neural operators: DeepONet merges the branch and trunk outputs through an inner product to produce the final solution prediction.For one-dimensional outputs, both subnetworks produce w-dimensional vectors before their inner product is computed.
  • Neural operators: During inference, vanilla DeepONet uses a clean input-function dataset and may require interpolation because its input locations are fixed.The corresponding solution is obtained by a forward DeepONet pass at each required x location.

3. Modeling total uncertainty

The section defines total uncertainty through predictive distributions that combine data noise with uncertainty in neural-network parameters. Posterior samples enable predictive PDFs or CDFs, while Gaussian-mixture or fitted Gaussian representations provide practical approximations and separate aleatoric from epistemic components.

  • Uncertainty modeling: Total uncertainty is modeled with predictive distributions and first- and second-order statistics for confidence intervals and uncertainty summaries.The framework integrates these quantities into scientific machine learning methods.
  • Predictive distributions: The predictive distribution p(u|x, D) is obtained by integrating over the posterior distribution of neural-network parameters.This Bayesian model average represents uncertainty over plausible parameter values given the data.
  • Posterior inference: Standard neural-network training uses a point estimate of parameters and therefore does not account for epistemic uncertainty.Approximate inference instead approximates the posterior distribution or obtains samples from it.
  • Uncertainty decomposition: The predictive variance separates approximate total uncertainty into aleatoric and epistemic components.The aleatoric component represents data noise, while the epistemic component arises from uncertainty represented by posterior parameter samples.
  • Practical evaluation: In practice, Gaussian or Student-t distributions can be fitted to posterior solution samples to evaluate predictive PDFs and CDFs.The fitted Gaussian uses the sample-based mean and variance at each location x.
  • Predictive distributions: Posterior samples produce a predictive PDF by combining the conditional distributions p(u|x, θ̂j) across sampled parameter values.With Gaussian conditional distributions, the resulting predictive distribution is a Gaussian mixture.
  • Noise modeling: Heteroscedastic noise can be modeled by making the variance location-dependent and including it as part of the neural-network output.Under this extension, each posterior sample can have both a different mean and a different variance.
  • Scientific machine learning: For mixed PDE datasets, separate likelihood functions are used for f, b, u, and λ observations, whose product forms the total likelihood.Noise scales may differ across data categories and can be known or learned during optimization.

4. Methods for uncertainty quantification

The paper integrates Bayesian inference, ensembles, functional priors, and stochastic-process methods into a unified SciML uncertainty-quantification framework. Posterior or ensemble parameter samples support approximate Bayesian model averaging, while extensions address historical data, stochastic PDEs, and neural operators.

  • Bayesian and ensemble inference: Posterior parameter samples are combined through approximate Bayesian model averaging to produce predictive distributions and first- and second-order uncertainty statistics.The framework first obtains samples of NN parameters and then uses them in approximate BMAs.
  • Bayesian and ensemble inference: Bayesian methods approximate p(θ|D) explicitly or implicitly, including MCMC, variational inference, and Laplace approximation.MCMC samples from the posterior, whereas VI uses an optimized variational distribution; the paper reviews these methods for SciML.
  • SciML integration: The framework combines these sampling methods with PINNs and DeepONets for learning solutions of PDEs, stochastic PDEs, and neural operator mappings.The method overview distinguishes parameter-sampling techniques from their combinations with PINNs and DeepONets.
  • Bayesian and ensemble inference: Ensembles obtain multiple posterior modes or loss minima and reuse their parameter settings as samples in approximate Bayesian model averaging.Deep ensembles train independently initialized networks, snapshot ensembles collect local optima during one trajectory, and SWAG fits a Gaussian to those optima.
  • Functional priors: Historical data can define functional priors through a pretrained GAN whose parameter distribution matches the historical data distribution.GAN-FP uses the generator output as a functional prior and can subsequently apply posterior inference.
  • Functional priors: Physics-agnostic BNN-FP and GAN-FP infer parameters from new noisy function data after pretraining the DeepONet with noiseless data.These approaches are presented as novel contributions for handling noisy data during inference.
  • Stochastic PDEs: The paper extends prior techniques to stochastic processes and SPDEs, including GP+PI-GAN for deterministic PDEs with noisy source data.GP+PI-GAN samples source terms from a GP before fitting PI-GAN, producing solution statistics that include epistemic uncertainty from limited and noisy data.
  • Stochastic PDEs: U-NNPC+ adds a decoder network specific to f to extend uncertainty modeling beyond the original U-NNPC formulation.The additional decoder receives a low-dimensional encoding of f and outputs a reconstructed source quantity.

5. Accuracy and uncertainty quality evaluation and improvement

The paper evaluates predictive accuracy, likelihood, calibration, sharpness, and dispersion using metrics with different data requirements, then applies post-training calibration to improve uncertainty predictions. Calibration can be adapted across function approximation, PDE, SPDE, and neural-operator settings.

  • Evaluation metrics: The evaluation framework supports model selection, UQ-method comparison, and overall quality assessment using metrics applied to statistics or the full predictive distribution.Metrics may use first- or second-order statistics, or the predictive PDF or CDF.
  • Accuracy and predictive capacity: Bayesian model averaging can improve mean-prediction accuracy while also providing predictive uncertainty distributions.The mean is approximated by averaging predictions from sampled NN parameters and is evaluated with relative ℓ2 error.
  • Evaluation metrics: MPL measures predictive capacity, RL2E measures mean-prediction accuracy, and RMSCE measures statistical consistency between predictions and data.PIW and SDCV provide secondary measures of prediction sharpness and uncertainty dispersion.
  • Accuracy and predictive capacity: High MPL values indicate that validation or test data are highly probable under the predictive distribution.MPL evaluates the PDF value assigned to tested datapoints under a Gaussian predictive approximation.
  • Calibration: Calibration assesses whether predicted statistical intervals contain the expected proportion of observed datapoints.Calibration plots compare expected and observed proportions, although each x location may have at most one available sample in practice.
  • Calibration: Post-training calibration uses a left-out calibration dataset and can also address model misspecification, such as an incorrect noise model.The paper presents three approaches for improving calibration after training.
  • Calibration: Variance scaling calibrates Gaussian predictions by replacing N(μ̄(x), σ̄²(x)) with N(μ̄(x), s²σ̄²(x)) without changing the mean.The scale factor is optimized using a calibration-related metric.
  • Calibration: A CDF-based calibration approach modifies both the predicted mean and variance by applying an estimated calibration map to predictive CDFs.The map is estimated from calibration data when the predictive model is miscalibrated.

6. Comparative study

The comparative study evaluates UQ methods across function approximation, PDE, and operator-learning problems, showing trade-offs among accuracy, calibration, computational cost, and uncertainty behavior. Learned priors, calibration, and multiple metrics materially affect method comparisons.

  • Function approximation: GP, HMC, MCD, DEns, and SEns are the most competitive methods for the function-approximation problem, but MCD and SEns do not consistently increase uncertainty for OOD inputs.This distinction matters for risk-sensitive applications that must separate interpolation from extrapolation.
  • Evaluation metrics: Different metrics can rank methods differently: MCD outperforms HMC on MPL across noise and dataset cases, while HMC’s larger uncertainty is desirable when noise increases or data decrease.The study therefore argues that UQ comparisons require a plurality of evaluation metrics rather than a single score.
  • Function approximation: Epistemic uncertainty increases with greater NN expressivity, while learning the prior can partially reverse this effect and reduce the need for exhaustive architecture selection.The comparison varies widths and depths, linking larger trainable-parameter counts to greater epistemic uncertainty.
  • Calibration: Post-training calibration with a separate left-out dataset can substantially reduce RMSCE, whereas calibrating on training data typically overfits and can worsen RMSCE.Because calibration uses ID data, OOD performance may deteriorate; the supplied results specifically note this behavior for SWAG.
  • Function approximation: MFVI and LA can perform comparably with HMC when their priors are learned, although HMC remains the most calibrated method for unknown homoscedastic noise after cheaper-method calibration.In this setting, HMC, MFVI, and LA learn the prior and noise scale online before evaluating ID performance.
  • PDE and operator studies: Across broader experiments, lower-cost methods with learned priors can match more expensive methods, while HMC+FP often achieves the strongest comparisons and much smaller calibration error.For one mixed PDE study, HMC+FP’s RMSCE was at least six times smaller than that of the other methods, and its predictions were nearly calibrated.

7. Summary

The paper develops a unified UQ framework for SciML and evaluates it across function approximation, PDEs, and operator learning. Results show method-dependent trade-offs involving computational cost, calibration, accuracy, and out-of-distribution detection.

  • The framework combines uncertainty modeling, posterior inference, prior learning, and post-training calibration for neural PDEs and operators.
  • Functional priors learn heteroscedastic noise from historical data, while new methods address forward PDEs, mixed SPDEs, and noisy operator-learning inference.
  • Lower-cost methods such as MFVI and LA can achieve performance comparable to HMC and DEns, particularly with learned priors; calibration can significantly reduce calibration error.
  • GP+PI-GAN can outperform U-PINNs in some forward PDE cases but may become over-confident because it does not use the PDE when fitting source-term data.
  • PA-GAN-FP outperforms PA-BNN-FP for limited noisy operator-learning data, while U-DeepONet improves OOD mean predictions and its epistemic uncertainty supports OOD detection.
  • Future work should extend UQ methods to high-dimensional problems and large datasets, with scalability and cost-effectiveness remaining practical research directions.

A.2. Point estimates (standard neural network training)

Standard neural-network training produces deterministic point estimates through likelihood or posterior optimization. Bayesian extensions introduce priors and hyperparameters, but exact integration is generally infeasible and model misspecification can make Bayesian model averaging underperform.

  • Maximum likelihood estimation infers NN parameters by maximizing p(D|θ), and Gaussian likelihood yields mean squared error minimization.
  • Maximum a posteriori estimation combines likelihood and prior information, with a Gaussian prior producing ℓ2 regularization or weight decay.
  • Stochastic gradient descent updates MAP parameters using only a mini-batch, reducing per-iteration data usage while retaining a learning-rate-controlled update.
  • Unknown prior and noise parameters are treated as hyperparameters with hyperpriors, but exact integration is infeasible and type-II maximum likelihood is used as an approximation.
  • Bayesian model averaging is theoretically optimal under perfect model specification, including a suitable NN function space and correct data-noise model.
  • Model misspecification can cause Bayesian model averaging to perform worse than point estimates, including when the likelihood or prior is unsuitable.
  • Cold posterior tempering samples from p(θ|D)^(1/τ) with τ<1; more informed priors may remove the reported performance benefit.

B. Supplementary material to Section 4: Additional details of employed UQ methods

The supplementary material explains how predictive and sampling uncertainties are computed and how functional-prior and neural-network components are implemented. It emphasizes additive aleatoric and epistemic uncertainty, sampling trade-offs, and assumptions in GAN-based priors.

  • Gaussian-process regression combines data and covariance structure to produce predictive means and epistemic covariance, with σ²_uI added for total uncertainty.
  • In GP regression, higher observation-noise variance reduces information gained from data, leaving epistemic uncertainty increasingly determined by the prior.
  • MCMC constructs chains whose equilibrium distribution is p(θ|D), using burn-in and invariant transitions to obtain posterior samples.
  • Metropolis methods generate proposal states and accept or reject them, with the acceptance rule designed to preserve p(θ|D) as the invariant distribution.
  • HMC corrects discretization bias through Metropolis acceptance, but finite-step Hamiltonian simulation remains its main computational bottleneck.
  • Langevin dynamics is a cheaper HMC variant, while SGLD further reduces cost by combining Langevin dynamics with mini-batch stochastic gradients.
  • Functional-prior GAN training uses a gradient penalty coefficient ζ=10 and synthetically contaminates noiseless generated functions to match historical noise.
  • NNPC+ uses a decoder to compare PDE outputs with reconstructed source-term data after PCA, but the added expressivity may require regularization.

C.1.1. Secondary metrics evaluating statistical consistency

The paper supplements calibration with sharpness, dispersion, gold-standard agreement, and post-hoc calibration metrics. These measures distinguish statistically consistent predictions from uncertainty estimates that are merely broad or homogeneous.

  • Sharpness evaluates the concentration of predictive distributions independently of observed data, complementing calibration assessment.
  • SDCV measures dispersion in predicted standard deviations across locations, reflecting whether uncertainty varies across the domain.The paper motivates greater dispersion as potentially improving robustness away from training data.
  • For equal calibration, preferred models have smaller PIW and larger SDCV values, because narrow intervals and dispersed uncertainty are jointly favored.
  • NIP-G compares predicted variances with a gold standard on a 0-to-1 scale, while KL-G compares the corresponding predictive distributions more comprehensively.
  • Applying the calibration function Q to predictive CDFs produces a perfectly calibrated model, with Q estimated from calibration data and approximated using monotonic regression.The transformation changes both predicted means and variances.
  • An alternative calibration method fits an empirical distribution to sorted scaled residuals and combines it with predicted means and standard deviations.This approach also changes the predicted mean and variance at each location.

D.2. Additional results for the function approximation problem of Section 6.1

The appendix adds function-approximation results for known homoscedastic noise and unknown Student-t heteroscedastic noise, including OOD evaluation and calibration analyses.

  • Additional results cover known homoscedastic noise and unknown Student-t heteroscedastic noise cases.
  • The experiments evaluate methods on out-of-distribution data and provide post-training calibration results.
  • The appendix also examines MCD across different noise scales and dataset sizes.

D.2.1. Known homoscedastic noise and post-training calibration

For function approximation, the appendix compares uncertainty methods under known homoscedastic noise and examines calibration, noise misspecification, and posterior tempering.

  • Known homoscedastic noise and post-training calibration: Post-training calibration results compare GP, HMC, LD, MFVI, MCD, LA, DEns, SEns, and SWAG using mean and total-uncertainty predictions.Figures report uncalibrated and calibrated predictions, calibration plots, and RMSCE values.
  • Known homoscedastic noise and post-training calibration: GP and HMC are expected to be the most accurate methods, but final performance depends on the dataset and individual tuning.Large OOD uncertainties may worsen MPL because conservative predictions can be penalized.
  • Known homoscedastic noise and post-training calibration: MCD uncertainty predictions are examined across noise scales σu = 0.1, 0.3, 0.5 and dataset sizes N = 32, 16.
  • Known homoscedastic noise and post-training calibration: The unknown Student-t heteroscedastic setting uses a Gaussian likelihood, making it a model-misspecification case for h-HMC and h-MFVI.Results compare predictions with and without posterior tempering.
  • Known homoscedastic noise and post-training calibration: In the misspecified case, changing posterior temperature controls conservativeness, and a cold posterior performs better in predictive capacity measured by MPL.The figures show mean, total uncertainty, and predicted aleatoric uncertainty for h-HMC and h-MFVI.

D.3. Additional results for the mixed PDE problem of Section 6.2

Additional mixed-PDE experiments test standard, large-noise, extrapolation, and steep-boundary-layer settings, revealing satisfactory predictions alongside localized over-confidence.

  • The appendix adds mixed-PDE results for MFVI, MCD, larger noise, extrapolation, and steep boundary layers.
  • In the standard case, predictions for u and f are satisfactory, but λ predictions are over-confident in some domain regions.
  • Increasing the data-noise scale from 0.05 to 0.1 produces larger predicted total uncertainty in the HMC results.

D.3.3. Extrapolation case: data concentrated on one side of the space domain

The extrapolation experiment shows that HMC can infer λ beyond the training-data region by leveraging its PDE coupling with u and f, with computational errors generally covered by 95% CIs. Additional cases compare uncertainty predictions across methods, noise levels, and boundary-layer difficulty.

  • Extrapolation case: HMC accurately predicts λ in x ∈ [−1, 0] despite having no training data there.The passage attributes this to λ being coupled with u and f through the PDE.
  • Extrapolation case: Information from u and f is leveraged through the PDE to support inference of λ in the extrapolation region.
  • Extrapolation case: Computational errors for u, f, and λ remain within the 95% CIs in the extrapolation case.
  • Steep boundary layers case: In the steep-boundary-layer case, HMC keeps computational errors for u, f, and λ within 95% CIs across most domains.Small left-side fluctuations in f may arise from the steep boundary layer, while abundant right-boundary data coincide with no observed right-side fluctuations.
  • Comparative uncertainty cases: The appendix compares MFVI, MCD, and HMC using epistemic and total uncertainty predictions under standard, large-noise, extrapolation, and steep-boundary-layer cases.The stated aleatoric noise scales include σu = 0.052 for MFVI, σu = 0.05 for MCD, σu = 0.1 for large-noise HMC, and σu = 0.05 for extrapolation and steep-boundary-layer HMC.

D.4. Additional results for the mixed stochastic problem of Section 6.3

The mixed stochastic results evaluate post-training calibration using held-out noisy realizations of u and λ. The experiments examine how calibration approaches and the number of left-out realizations affect RMSCE.

  • Calibration metric: RMSCE is computed from ensemble prediction realizations for clean test realizations of u sampled at 101 points.The procedure generates 100 realizations for each of U-PI-GAN, U-NNPC, and U-NNPC+, yielding 300 realizations.
  • Calibration setup: Post-training calibration is evaluated using 2–50 realizations of u and λ withheld from training.Calibration uses prediction realizations sampled at the training-data locations.
  • Calibration results: Even 2–10 left-out noisy realizations can reduce calibration error by half in some cases.The experiment also compares calibration methods and observes a decreasing RMSCE trend as calibration-dataset size increases.

D.5. Additional results for the forward PDE problem of Section 6.4

For the forward PDE problem with only six f datapoints and no boundary measurements, GP+PI-GAN and U-PINN produce different uncertainty behavior. Coupling through the PDE gives U-PINN lower uncertainty for f at the boundaries and avoids GP+PI-GAN’s underestimated uncertainty for u.

  • Limited f data: With Nf = 6 and no f measurements at the two boundaries, GP+PI-GAN predicts larger boundary uncertainty for f than U-PINN.The passage attributes this to GP+PI-GAN not using the PDE during training.
  • Uncertainty coupling: GP+PI-GAN underestimates the uncertainty of u compared with U-PINN when the PDE does not couple their approximations during training.
  • λ data model: For positive λ, the noisy observations are modeled with a log-normal distribution whose noise is proportional to the clean value λc(x, y).The transformed quantity λ̃ = log λ is modeled with a Gaussian distribution.
  • DeepONet input limitation: The input-dependent standardization of λ̃ restricts λθ evaluation to the original (x, y) training grid.The stated alternatives are avoiding standardization or using shared standardization across points.

D.7. Additional results for the operator learning problem of Section 6.5

Additional operator-learning results show substantial degradation on OOD inference data, with PA-BNN-FP generally better calibrated than PA-GAN-FP. Post-training calibration using a few held-out noisy datapoints can substantially reduce RMSCE in some cases.

  • OOD inference: On OOD data, PA-BNN-FP and PA-GAN-FP have RL2E values close to 40–50%.The computational errors are mostly covered by PA-BNN-FP’s 95% CIs but not by PA-GAN-FP’s.
  • Inference experiment: The operator-learning experiments use partial noisy measurements of λ and u to reconstruct both fields, including samples generated from a Gaussian process with l = 0.2.
  • Post-training calibration: Post-training calibration with 2–10 left-out noisy datapoints can halve calibration error in some cases, especially for OOD data with variance re-weighting.
  • OOD uncertainty: PA-BNN-FP is more calibrated than PA-GAN-FP on OOD data, while PA-GAN-FP produces confident predictions for unseen λ values.
  • Implementation details: The appendix provides hyperparameter glossaries and method-specific settings for the comparative study, including optimizer and sampling parameters.

D.9. Neural network architectures used in the comparative study

The comparative study uses mostly standardized neural-network architectures across function approximation, mixed PDE, forward PDE, and operator-learning problems, with specialized architectures for GANs, NNPC, and DeepONets.

  • Function approximation uses neural networks with 2 hidden layers, 50 neurons per layer, hyperbolic-tangent activations, and standard-normal weight and bias priors.
  • Mixed PDE experiments compare six UQ methods and use two similarly structured neural networks as surrogates for the solution u and reaction rate λ.
  • Forward PDE and operator-learning experiments reuse the function-approximation BNN architecture and priors, while separate architectures are reported for GANs, PI-GANs, NNPC, and neural operators.
  • GAN and PI-GAN configurations specify hidden-layer width, depth, training steps, and K, the dimension of θ and output dimension of each generator network when two are used.
  • For selected experiments, two generator networks separately take Gaussian noise and coordinates to improve computational efficiency in posterior estimation.
  • NNPC and NNPC+ use separate neural networks for the mean and modes of each u and λ output.
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