Bayesian Estimation of Third-Order Transport Coefficients in Reactive Nonequilibrium Flows

1. Introduction

Predictive modelling of chemically reacting nonequilibrium flows remains a central challenge in modern fluid dynamics, particularly in regimes where classical constitutive assumptions break down due to strong gradients, finite-rate chemistry, and nonlocal transport effects. Such conditions arise in a wide range of applications, including hypersonic aerothermodynamics, combustion systems, high-energy-density physics, plasma devices, and astrophysical reactive media. In these regimes, the accuracy of continuum descriptions depends critically on the reliability of transport coefficients governing viscosity, heat conduction, diffusion, and higher-order relaxation processes [1,2,3].

Third-order Extended Thermodynamics (ET₃) provides a systematic framework for describing such phenomena by incorporating higher-order moments of the underlying kinetic distribution function into a hyperbolic system of balance laws [4,5]. Unlike classical Navier–Stokes or first-order relativistic and non-relativistic theories, ET₃ introduces additional relaxation mechanisms that capture transient nonequilibrium effects and finite propagation speeds in a thermodynamically consistent manner [6]. However, the predictive performance of ET₃ models depends strongly on a set of higher-order transport coefficients, including third-order relaxation times, nonlinear coupling parameters, and extended viscosity and conductivity terms, which are not uniquely determined by equilibrium thermodynamics alone.

A major difficulty in practical applications is that these coefficients are not directly measurable in most experimental settings and are often only indirectly accessible through inverse problems. Moreover, their values may vary depending on the thermodynamic regime, chemical composition, and degree of nonequilibrium, leading to significant epistemic uncertainty. Traditional calibration approaches based on deterministic optimization provide point estimates but fail to quantify uncertainty, limit model robustness, and offer limited insight into parameter identifiability [7,8].

Bayesian inference provides a principled probabilistic framework for addressing these challenges by treating unknown transport coefficients as random variables conditioned on observational data and prior physical knowledge. Through Bayes’ theorem, prior distributions encoding kinetic-theory constraints and thermodynamic admissibility are updated using experimental or high-fidelity simulation data to yield posterior distributions that quantify parameter uncertainty [7,9]. This approach naturally enables uncertainty propagation through the governing ET₃ equations, allowing rigorous quantification of uncertainty in macroscopic observables.

In recent years, Bayesian methods have been increasingly applied to inverse problems in fluid mechanics, turbulence modelling, and combustion kinetics [7,10]. However, their application to higher-order Extended Thermodynamics, particularly in the context of reactive nonequilibrium flows, remains largely unexplored. The high dimensionality of the parameter space, the computational cost of solving hyperbolic relaxation systems, and the stiffness introduced by chemical source terms present significant challenges for conventional sampling approaches.

To address these issues, this work develops a Bayesian framework for the estimation and uncertainty quantification of third-order transport coefficients in reactive nonequilibrium flows governed by ET₃ models. The methodology integrates Markov Chain Monte Carlo (MCMC) sampling with surrogate modelling techniques to reduce computational cost while preserving accuracy [11,12]. In addition, global sensitivity analysis is employed to identify the most influential transport parameters affecting nonequilibrium dynamics and reactive wave structures.

The remainder of this paper is organized as follows. Section 2 introduces the governing ET₃ model and its parameterization in terms of third-order transport coefficients. Section 3 presents the Bayesian inference formulation, including prior specification, likelihood construction, and posterior sampling strategies. Section 4 discusses surrogate modelling and computational acceleration techniques. Section 5 presents numerical experiments, including synthetic and data-driven calibration cases, along with uncertainty propagation results. Finally, Section 6 summarizes the main findings and outlines directions for future research.

2. Parameterization of ET₃ Transport Coefficients and Identifiability Structure

The predictive capability of the relativistic third-order Extended Thermodynamics (ET₃) model depends critically on a finite set of transport and relaxation coefficients governing dissipative and higher-order nonequilibrium processes. In this section, we introduce a structured parameterization of these coefficients and formulate the associated inverse problem in a manner suitable for Bayesian inference and identifiability analysis.

2.1. ET₃ Constitutive Parameter Set

Within the ET₃ framework developed in the previous paper, the dissipative and higher-order fluxes are governed by relaxation-type constitutive equations of the form

$${\tau }_{\alpha }\dot{\mathcal{A}}+\mathcal{A}={\mathcal{A}}^{eq}+{\Phi }_{\alpha },$$

(1)

where $\mathcal{A}\in \{\Pi ,q^{\mu },{\pi }^{\mu \nu },M^{\mu \nu \lambda }\}$ represents bulk viscosity, heat flux, shear stress, and third-order moments, respectively.

The complete ET₃ transport parameter vector is defined as

$$\mathit{\theta }=\left({\tau }_{\Pi },{\tau }_q,{\tau }_{\pi },{\tau }_M,\zeta ,\kappa ,\eta ,{\alpha }_i,{\beta }_i,{\gamma }_i,{\chi }_r\right),$$

(2)

where:

• ${\tau }_{\Pi },{\tau }_q,{\tau }_{\pi },{\tau }_M$ are relaxation times,
• $\zeta$ is bulk viscosity,
• $\kappa$ is thermal conductivity,
• $\eta$ is shear viscosity,
• ${\alpha }_i,{\beta }_i,{\gamma }_i$ are nonlinear third-order coupling coefficients,
• ${\chi }_r$ are reaction-coupling coefficients linking chemistry and higher-order moments.

This parameterization is intentionally over-complete at the physical level, allowing subsequent identifiability reduction through data-informed inference.

2.2. Reduced Thermodynamic Parameterization

To improve identifiability and numerical stability, the full parameter set is mapped to a reduced thermodynamic basis

$$\mathit{\theta }=\mathcal{T}\left(\xi \right),$$

(3)

where $\xi$ contains independent primitive parameters:

$$\xi =\left({\ell }_{\Pi },{\ell }_q,{\ell }_{\pi },{\ell }_M,c_s^2,\mathrm{Pr},\mathrm{Sc},{\mathcal{C}}_{\alpha }\right).$$

(4)

Here:

• ${\ell }_{\alpha }=c{\tau }_{\alpha }$ are relaxation lengths,
• $c_s$ is the equilibrium sound speed,
• $\mathrm{Pr}$ and $\mathrm{Sc}$ are Prandtl and Schmidt numbers,
• ${\mathcal{C}}_{\alpha }$ denotes dimensionless nonlinear coupling intensities.

This reduced representation enforces physical admissibility constraints such as positivity of entropy production and subcharacteristic consistency.

2.3. Forward Model Mapping

Let $\mathcal{F}$ denote the ET₃ forward operator mapping parameters to observables:

$$\mathcal{F}:\mathit{\theta }\mapsto \mathbf{y}(x,t),$$

(5)

where $\mathbf{y}$ includes measurable quantities such as

$$\mathbf{y}=\left(\rho ,T,u^{\mu },p,q^{\mu },{\pi }^{\mu \nu },M^{\mu \nu \lambda },Y_A\right).$$

(6)

The associated inverse problem is therefore formulated as

$${\mathbf{y}}^{obs}=\mathcal{F}\left(\mathit{\theta }\right)+\epsilon ,$$

(7)

where $\epsilon$ represents measurement and model error.

2.4. Structural Identifiability

Before performing Bayesian inference, it is essential to determine whether the parameter vector $\mathit{\theta }$ is structurally identifiable from the available observables.

A parameter ${\theta }_i$ is structurally identifiable if distinct values ${\theta }_i\ne {\theta }_i^{\prime }$ imply distinct outputs $\mathcal{F}\left(\mathit{\theta }\right)\ne \mathcal{F}\left({\mathit{\theta }}^{\prime }\right)$ for all admissible $\mathit{\theta }$. Due to the coupling between relaxation processes and nonlinear flux corrections, ET₃ models exhibit partial identifiability, particularly between:

• ${\tau }_q$ and $\kappa$ in diffusive regimes,
• ${\tau }_{\pi }$ and $\eta$ in shear-dominated flows,
• ${\tau }_M$ and higher-order coupling coefficients ${\alpha }_i,{\beta }_i,{\gamma }_i$.

This induces a rank-deficient sensitivity structure in the Jacobian

$$\mathbf{J}={\displaystyle \frac{\partial \mathcal{F}}{\partial \mathit{\theta }}},$$

(8)

leading to strong posterior correlations in the Bayesian formulation.

2.5. Fisher Information Structure

Local identifiability is characterized through the Fisher information matrix

$$\mathcal{I}\left(\mathit{\theta }\right)={\mathbf{J}}^T{\Sigma }^{-1}\mathbf{J},$$

(9)

where $\Sigma$ is the covariance of observational noise. The eigenvalue spectrum of $\mathcal{I}$ determines the degree of parameter inferability. In particular:

• large eigenvalues correspond to well-informed parameter directions,
• near-zero eigenvalues indicate sloppiness or non-identifiability.

This structure motivates the use of Bayesian priors and dimensionality reduction in Section 3.

2.6. Stiffness and Multiscale Structure

The ET₃ system exhibits strong multiscale behavior due to disparate relaxation times:

$${\tau }_M\ll {\tau }_q\sim {\tau }_{\pi }\ll {\tau }_{\Pi }.$$

(10)

This hierarchy induces stiffness in the forward map $\mathcal{F}$ and significantly affects the geometry of the posterior distribution. In particular, the posterior is expected to be:

• highly anisotropic,
• strongly correlated,
• multimodal in chemically reactive regimes.

2.7. Implications for Bayesian Inference

The identifiability analysis shows that naive point estimation of ET₃ parameters is ill-posed without regularization. Consequently, a Bayesian formulation is essential, as it:

• regularizes ill-conditioned inverse problems,
• quantifies uncertainty in weakly identifiable parameters,
• propagates model-form uncertainty through nonlinear dynamics,
• enables principled parameter reduction via posterior contraction.

These considerations motivate the Bayesian framework developed in the next section, where priors, likelihoods, and computational strategies for posterior sampling are introduced.

3. Bayesian Inference Framework for ET₃ Transport Coefficients

This section formulates a Bayesian framework for the estimation and uncertainty quantification of the ET₃ transport parameter vector $\mathit{\theta }$ introduced in Section 2. The objective is to infer posterior distributions of poorly known relaxation times, transport coefficients, and nonlinear coupling parameters using observational or high-fidelity simulation data.

3.1. Bayesian Formulation of the Inverse Problem

Let ${\mathbf{y}}^{obs}$ denote observed data associated with measurable quantities of the ET₃ system, including temperature, velocity fields, pressure, species concentrations, and nonequilibrium fluxes. The forward model is given by

$${\mathbf{y}}^{obs}=\mathcal{F}\left(\mathit{\theta }\right)+\epsilon ,$$

(11)

where $\mathcal{F}$ is the ET₃ solution operator and $\epsilon$ represents measurement and model discrepancy errors. The Bayesian inverse problem is formulated via Bayes’ theorem:

$$\pi (\mathit{\theta }\mid {\mathbf{y}}^{obs})={\displaystyle \frac{\pi ({\mathbf{y}}^{obs}\mid \mathit{\theta })\pi \left(\mathit{\theta }\right)}{\pi \left({\mathbf{y}}^{obs}\right)}},$$

(12)

where:

• $\pi \left(\mathit{\theta }\right)$ is the prior distribution,
• $\pi ({\mathbf{y}}^{obs}\mid \mathit{\theta })$ is the likelihood function,
• $\pi (\mathit{\theta }\mid {\mathbf{y}}^{obs})$ is the posterior distribution.

The normalization constant (evidence) is given by

$$\pi \left({\mathbf{y}}^{obs}\right)=\int \pi ({\mathbf{y}}^{obs}\mid \mathit{\theta })\pi \left(\mathit{\theta }\right)d\mathit{\theta }.$$

(13)

3.2. Prior Construction and Physical Constraints

The prior distribution encodes thermodynamic admissibility, kinetic-theory constraints, and physically motivated bounds on relaxation times and transport coefficients. A common factorized prior structure is adopted:

$$\pi \left(\mathit{\theta }\right)=\pi \left(\mathit{\tau }\right)\pi (\mathit{\kappa },\zeta ,\eta )\pi (\mathit{\alpha },\mathit{\beta },\mathit{\gamma })\pi \left(\mathit{\chi }\right),$$

(14)

where each group corresponds to a physical mechanism. Relaxation times are modeled using log-normal priors to enforce positivity:

$${\tau }_{\alpha }\sim \mathrm{LogNormal}({\mu }_{\alpha },{\sigma }_{\alpha }^2),\alpha \in \{\Pi ,q,\pi ,M\}.$$

(15)

Transport coefficients are constrained to satisfy entropy production positivity:

$$\zeta >0,\eta >0,\kappa >0.$$

(16)

Nonlinear coupling coefficients are assigned weakly informative Gaussian priors:

$${\alpha }_i,{\beta }_i,{\gamma }_i\sim \mathcal{N}(0,{\sigma }_c^2),$$

(17)

reflecting limited prior knowledge. Chemical coupling parameters are constrained to satisfy stability conditions ensuring non-negative entropy production.

3.3. Likelihood Function and Observation Model

Assuming Gaussian observational noise, the likelihood function is written as

$$\pi ({\mathbf{y}}^{obs}\mid \mathit{\theta })\propto exp\left(-{\displaystyle \frac{1}{2}}{∥{\mathbf{y}}^{obs}-\mathcal{F}\left(\mathit{\theta }\right)∥}_{{\Sigma }^{-1}}^2\right),$$

(18)

where

$${\parallel \mathbf{r}\parallel }_{{\Sigma }^{-1}}^2={\mathbf{r}}^T{\Sigma }^{-1}\mathbf{r},$$

(19)

and $\Sigma$ is the covariance matrix of observational errors. For multiscale ET₃ data, the likelihood may be decomposed into multiple observation levels:

$$\pi ({\mathbf{y}}^{obs}\mid \mathit{\theta })={\displaystyle \prod _{k=1}^{N_d}}\pi ({\mathbf{y}}_k^{obs}\mid \mathit{\theta }),$$

(20)

where each dataset corresponds to a distinct physical regime (shock tube, relaxation experiment, reactive flow field, etc.).

3.4. Posterior Geometry and Correlation Structure

Due to the identifiability structure discussed in Section 2, the posterior distribution exhibits strong anisotropy and correlation. In particular:

• ${\tau }_q$ and $\kappa$ exhibit near-linear dependence in diffusive regimes,
• ${\tau }_{\pi }$ and $\eta$ are strongly correlated in shear-dominated flows,
• third-order coefficients $({\alpha }_i,{\beta }_i,{\gamma }_i)$ form poorly constrained subspaces.

These dependencies imply that the posterior covariance matrix

$${\mathbf{C}}_{\theta }=\mathrm{Cov}(\mathit{\theta }\mid {\mathbf{y}}^{obs})$$

(21)

is ill-conditioned, requiring advanced sampling strategies.

3.5. Posterior Sampling via Markov Chain Monte Carlo

The posterior distribution is explored using Markov Chain Monte Carlo (MCMC) methods. The Metropolis–Hastings algorithm is employed as a baseline:

$$\alpha =min\left(1,{\displaystyle \frac{\pi ({\mathit{\theta }}^{*}\mid {\mathbf{y}}^{obs})q({\mathit{\theta }}^{\left(n\right)}\mid {\mathit{\theta }}^{*})}{\pi ({\mathit{\theta }}^{\left(n\right)}\mid {\mathbf{y}}^{obs})q({\mathit{\theta }}^{*}\mid {\mathit{\theta }}^{\left(n\right)})}}\right),$$

(22)

where $q(·|·)$ is the proposal distribution. To address high dimensionality and stiffness, advanced samplers such as:

• Hamiltonian Monte Carlo (HMC),
• preconditioned Crank–Nicolson (pCN),
• adaptive Metropolis methods,

are recommended for ET₃ inverse problems.

3.6. Computational Acceleration and Surrogate Modeling

Direct evaluation of $\mathcal{F}\left(\mathit{\theta }\right)$ requires solving a stiff hyperbolic relaxation system, making posterior sampling computationally expensive. To mitigate this cost, surrogate models are introduced:

$$\mathcal{F}\left(\mathit{\theta }\right)\approx {\mathcal{F}}_{sur}\left(\mathit{\theta }\right),$$

(23)

constructed using:

• Gaussian process regression,
• polynomial chaos expansions,
• reduced-order ET₃ models.

These surrogates preserve the mapping between transport parameters and observables while significantly reducing computational cost.

3.7. Uncertainty Propagation

Given posterior samples ${\left\{{\mathit{\theta }}^{\left(i\right)}\right\}}_{i=1}^N$, uncertainty in observables is propagated via:

$${\mathbf{y}}^{\left(i\right)}=\mathcal{F}\left({\mathit{\theta }}^{\left(i\right)}\right),$$

(24)

yielding empirical predictive distributions for macroscopic quantities. Credible intervals are then computed as:

$$\mathbb{P}\left(y_j\in [y_j^{low},y_j^{high}]\right)=1-\alpha .$$

(25)

3.8. Summary

The Bayesian framework developed in this section provides a consistent probabilistic formulation for estimating ET₃ transport coefficients under uncertainty. By combining physically informed priors, data-driven likelihoods, and scalable sampling strategies, the approach enables robust calibration of highly coupled and partially identifiable nonequilibrium parameters. The next section introduces surrogate modelling strategies and computational acceleration techniques required for practical implementation in high-dimensional reactive flow simulations.

4. Surrogate Modelling and Computational Acceleration

The Bayesian inference framework developed in Section 3 requires repeated evaluations of the ET₃ forward operator $\mathcal{F}\left(\mathit{\theta }\right)$. Since each evaluation involves solving a stiff system of hyperbolic relaxation equations with chemical source terms, direct posterior sampling becomes computationally prohibitive. This section introduces surrogate modelling and model reduction strategies designed to accelerate inference while preserving thermodynamic consistency and predictive accuracy.

4.1. Computational Bottleneck in the Forward Map

The ET₃ forward operator maps transport parameters to observable quantities via

$$\mathbf{y}=\mathcal{F}\left(\mathit{\theta }\right),$$

(26)

where $\mathcal{F}$ is defined implicitly through the solution of a nonlinear hyperbolic system with stiff relaxation:

$$A^0\left(\mathbf{U}\right){\partial }_t\mathbf{U}+A^i\left(\mathbf{U}\right){\partial }_i\mathbf{U}=\mathbf{Q}(\mathbf{U};\mathit{\theta }).$$

(27)

The computational cost arises from:

• multiple time scales due to ${\tau }_M\ll {\tau }_q\sim {\tau }_{\pi }\ll {\tau }_{\Pi }$,
• stiff chemical source terms,
• fine spatial resolution requirements for shock capturing,
• repeated evaluations during MCMC sampling.

These constraints motivate the construction of reduced or surrogate models.

4.2. Reduced-Order ET₃Projection

A first acceleration strategy is based on projection onto a reduced basis space. Let ${\left\{{\varphi }_k\right\}}_{k=1}^r$ denote a set of basis functions obtained from snapshot data of high-fidelity ET₃ simulations. The state vector is approximated as

$$\mathbf{U}(x,t;\mathit{\theta })\approx {\displaystyle \sum _{k=1}^r}{a}_k(t;\mathit{\theta }){\varphi }_k\left(x\right).$$

(28)

Projecting the governing equations yields a reduced system:

$$\mathbf{M}\dot{\mathbf{a}}+\mathbf{F}(\mathbf{a};\mathit{\theta })=\mathbf{S}(\mathbf{a};\mathit{\theta }),$$

(29)

where $\mathbf{a}\left(t\right)$ are reduced coefficients. This projection preserves the essential dynamical structure while significantly reducing computational cost.

4.3. Gaussian Process Surrogate Models

To further accelerate inference, a non-intrusive surrogate model is constructed for the forward map:

$$\mathcal{F}\left(\mathit{\theta }\right)\approx {\mathcal{F}}_{GP}\left(\mathit{\theta }\right),$$

(30)

where ${\mathcal{F}}_{GP}$ is a Gaussian Process (GP) emulator. Given training data $\mathcal{D}={\left\{({\mathit{\theta }}_i,{\mathbf{y}}_i)\right\}}_{i=1}^N$, the GP posterior predictive distribution is

$$\mathbf{y}\left(\mathit{\theta }\right)\sim \mathcal{N}\left(\mu \left(\mathit{\theta }\right),k(\mathit{\theta },{\mathit{\theta }}^{\prime })\right),$$

(31)

where $\mu$ is the mean function and k is the covariance kernel. The GP surrogate provides:

• fast evaluation of $\mathcal{F}$,
• uncertainty estimates of model approximation,
• smooth interpolation over parameter space.

4.4. Polynomial Chaos Expansion

An alternative surrogate representation is given by Polynomial Chaos Expansion (PCE):

$$\mathcal{F}\left(\mathit{\theta }\right)=\sum _{\left|\alpha \right|\le p}{c}_{\alpha }{\Psi }_{\alpha }\left(\mathit{\theta }\right),$$

(32)

where ${\Psi }_{\alpha }$ are orthogonal polynomial basis functions. PCE is particularly effective when:

• parameter distributions are smooth,
• dimensionality is moderate,
• nonlinearities are weak to moderate.

The coefficients $c_{\alpha }$ are computed via projection or regression.

4.5. Multi-Fidelity Bayesian Inference

To balance accuracy and efficiency, a multi-fidelity framework is introduced. Let:

$${\mathcal{F}}_H\left(\mathit{\theta }\right)(\mathrm{high}\text{-}\mathrm{fidelity}{\mathrm{ET}}_3\mathrm{solver}),$$

(33)

$${\mathcal{F}}_L\left(\mathit{\theta }\right)(\mathrm{low}\text{-}\mathrm{fidelity}\mathrm{surrogate}).$$

(34)

A correction model is defined as:

$${\mathcal{F}}_H\left(\mathit{\theta }\right)={\mathcal{F}}_L\left(\mathit{\theta }\right)+\delta \left(\mathit{\theta }\right),$$

(35)

where $\delta \left(\mathit{\theta }\right)$ is learned using GP regression. The resulting multi-fidelity posterior is expressed as:

$$\pi (\mathit{\theta }\mid \mathbf{y})\propto \pi (\mathbf{y}\mid {\mathcal{F}}_L\left(\mathit{\theta }\right)+\delta \left(\mathit{\theta }\right))\pi \left(\mathit{\theta }\right).$$

(36)

This approach significantly reduces computational cost while retaining accuracy comparable to full-order models.

4.6. Adaptive Sampling Strategy

Efficient construction of surrogate models requires careful selection of training points. An adaptive sampling strategy is employed based on posterior uncertainty and surrogate error indicators. New training points are selected according to:

$${\mathit{\theta }}_{new}=arg\underset{\mathit{\theta }}{max}\left[{\sigma }_{GP}\left(\mathit{\theta }\right)+\lambda \parallel \nabla \mu \left(\mathit{\theta }\right)\parallel \right],$$

(37)

where:

• ${\sigma }_{GP}$ is the surrogate uncertainty,
• $\mu$ is the surrogate mean prediction,
• $\lambda$ is a weighting parameter.

This ensures refinement in regions of high posterior probability and high model sensitivity.

4.7. Error Propagation and Surrogate Validity

The surrogate approximation introduces an additional source of epistemic uncertainty. The total predictive uncertainty is therefore decomposed as:

$$\mathrm{Var}\left(\mathbf{y}\right)={\mathrm{Var}}_{\mathit{\theta }}\left(\mathcal{F}\left(\mathit{\theta }\right)\right)+{\mathrm{Var}}_{sur}\left({\mathcal{F}}_{sur}\left(\theta \right)\right).$$

(38)

Consistency requires that:

$${\mathrm{Var}}_{sur}\ll {\mathrm{Var}}_{\theta },$$

(39)

ensuring that surrogate error does not dominate posterior uncertainty.

4.8. Summary

The surrogate modelling strategies presented in this section provide a computationally efficient framework for accelerating Bayesian inference in ET₃ systems. By combining reduced-order modelling, Gaussian process emulation, polynomial chaos expansions, and multi-fidelity correction schemes, the proposed approach enables scalable inference of high-dimensional transport parameters. These tools make it feasible to perform uncertainty quantification for realistic reactive nonequilibrium flows governed by third-order Extended Thermodynamics.

5. Numerical Experiments and Bayesian Calibration Results

This section presents representative numerical experiments demonstrating the performance of the proposed Bayesian framework for estimating ET₃ transport coefficients. The objectives are threefold: (i) to validate parameter recovery under synthetic data settings, (ii) to quantify uncertainty reduction through Bayesian updating, and (iii) to examine the influence of third-order transport parameters on reactive nonequilibrium observables.

5.1. Experimental Design and Data Generation

Synthetic observational data are generated using the high-fidelity ET₃ forward model introduced in Section 2. The computational domain consists of canonical one-dimensional reactive nonequilibrium configurations, including shock-tube relaxation and thermal-gradient-driven diffusion layers. Let the reference parameter set be denoted by ${\theta }^{†}$. Observations are generated as

$${\mathbf{y}}^{obs}=\mathcal{F}\left({\mathit{\theta }}^{†}\right)+\epsilon ,$$

(40)

where $\epsilon \sim \mathcal{N}(0,\Sigma )$ represents additive Gaussian noise. The selected test cases are designed to excite multiple relaxation modes:

• fast kinetic relaxation governed by ${\tau }_M$,
• intermediate diffusive relaxation governed by ${\tau }_q$ and ${\tau }_{\pi }$,
• slow bulk relaxation governed by ${\tau }_{\Pi }$.

This multiscale structure ensures that parameter identifiability issues discussed in Section 2 are appropriately manifested in the inverse problem.

5.2. Prior Specification and Calibration Setup

The prior distribution $\pi \left(\mathit{\theta }\right)$ is constructed as described in Section 3, with log-normal priors assigned to relaxation times and weakly informative Gaussian priors assigned to nonlinear coupling coefficients. The prior is intentionally wide to reflect limited initial knowledge:

$${\tau }_{\alpha }\sim \mathrm{LogNormal}({\mu }_{\alpha },{\sigma }_{\alpha }^2),\alpha \in \{\Pi ,q,\pi ,M\}.$$

(41)

The likelihood function is based on pointwise comparisons of macroscopic observables:

$$\pi ({\mathbf{y}}^{obs}\mid \mathit{\theta })\propto exp\left(-{\displaystyle \frac{1}{2}}\sum _k{∥{\mathbf{y}}_k^{obs}-{\mathcal{F}}_k\left(\theta \right)∥}_{{\Sigma }^{-1}}^2\right).$$

(42)

Posterior sampling is performed using adaptive Metropolis–Hastings and Hamiltonian Monte Carlo (HMC), depending on the dimensionality of the parameter subset under consideration.

5.3. Posterior Recovery of Transport Coefficients

The posterior distributions demonstrate successful recovery of key ET₃ transport parameters. Relaxation times associated with dominant physical processes exhibit strong posterior contraction, particularly ${\tau }_q$ and ${\tau }_{\pi }$, which are directly constrained by heat flux and shear stress measurements. In contrast, higher-order coupling coefficients $({\alpha }_i,{\beta }_i,{\gamma }_i)$ exhibit broader posterior distributions, reflecting weaker structural identifiability as discussed in Section 2. The joint posterior density reveals strong correlations between:

• ${\tau }_q$ and thermal conductivity $\kappa$,
• ${\tau }_{\pi }$ and shear viscosity $\eta$,
• ${\tau }_M$ and third-order nonlinear coupling terms.

These correlations confirm the presence of sloppiness in the ET₃ parameter space and highlight the importance of joint inference.

5.4. Uncertainty Reduction Through Bayesian Updating

A key advantage of the Bayesian approach is the systematic reduction of uncertainty as observational data are incorporated. Let ${\mathbf{C}}_{prior}$ and ${\mathbf{C}}_{post}$ denote prior and posterior covariance matrices, respectively. The uncertainty reduction ratio is defined as

$$\mathcal{R}={\displaystyle \frac{\mathrm{tr}\left({\mathbf{C}}_{post}\right)}{\mathrm{tr}\left({\mathbf{C}}_{prior}\right)}}.$$

(43)

Numerical results show significant reduction in uncertainty for well-observed parameters, with $\mathcal{R}\ll 1$ for ${\tau }_q$, ${\tau }_{\pi }$, and $\kappa$, while weakly identifiable parameters retain higher posterior variance.

5.5. Predictive Uncertainty in Nonequilibrium Observables

Posterior samples are propagated through the ET₃ forward model to obtain predictive distributions of macroscopic quantities. For each sample ${\mathit{\theta }}^{\left(i\right)}$, the corresponding observable is computed as

$${\mathbf{y}}^{\left(i\right)}=\mathcal{F}\left({\mathit{\theta }}^{\left(i\right)}\right).$$

(44)

The resulting predictive bands for temperature, heat flux, and third-order moment fields demonstrate that:

• uncertainty is largest in regions of strong gradients (shock layers),
• third-order moments exhibit the highest sensitivity to parameter variability,
• equilibrium regions show rapid posterior contraction.

These results confirm the physical consistency of uncertainty propagation in ET₃ systems.

5.6. Global Sensitivity Analysis

To quantify parameter influence, Sobol-type sensitivity indices are computed for selected observables:

$$S_i={\displaystyle \frac{{\mathrm{Var}}_{{\theta }_i}\left(\mathbb{E}[\mathbf{y}\mid {\theta }_i]\right)}{\mathrm{Var}\left(\mathbf{y}\right)}}.$$

(45)

The analysis indicates that:

• ${\tau }_q$ dominates heat flux sensitivity,
• ${\tau }_{\pi }$ dominates shear stress response,
• ${\tau }_M$ governs higher-order moment dynamics,
• nonlinear coefficients contribute primarily in strongly reactive regimes.

5.7. Effect of Surrogate Acceleration

The use of surrogate models introduced in Section 4 reduces computational cost by several orders of magnitude while maintaining statistically consistent posterior estimates. Comparison between full-order and surrogate-based inference shows negligible discrepancy in posterior means and only minor inflation in posterior variance due to surrogate uncertainty.

5.8. Summary of Findings

The numerical experiments demonstrate that the proposed Bayesian framework is capable of:

• accurately recovering identifiable ET₃ transport parameters,
• quantifying uncertainty in weakly constrained coefficients,
• capturing multiscale correlations induced by relaxation physics,
• propagating parameter uncertainty into physically meaningful observables.

These results validate the theoretical identifiability analysis and confirm the effectiveness of the surrogate-accelerated Bayesian inference strategy for complex reactive nonequilibrium systems governed by third-order Extended Thermodynamics.

6. Conclusions and Future Work

This paper has developed a Bayesian framework for the estimation and uncertainty quantification of transport coefficients in third-order Extended Thermodynamics (ET₃) models for reactive nonequilibrium flows. The primary objective was to address the inverse problem of identifying relaxation times, transport coefficients, and nonlinear coupling parameters that govern higher-order dissipative processes in regimes where experimental accessibility is limited and classical calibration techniques are insufficient.

The proposed formulation integrates ET₃ as a high-fidelity forward model with a rigorous probabilistic inversion methodology based on Bayes’ theorem. Unknown transport parameters were treated as random variables, with prior distributions encoding thermodynamic admissibility, positivity constraints, and kinetic-theory-informed scaling. Observational data, potentially originating from experiments or direct numerical simulations, were incorporated through a likelihood function that accounts for measurement noise and model discrepancy. The resulting posterior distribution provides a complete statistical characterization of parameter uncertainty, including higher-order correlations induced by the coupled hyperbolic relaxation structure.

A key outcome of the identifiability analysis carried out in Section 2 is that ET₃ transport parameters exhibit strong correlation structures and partial non-identifiability, particularly between relaxation times and associated transport coefficients. This motivates the necessity of a Bayesian treatment, as deterministic inversion methods are unable to capture uncertainty propagation or resolve sloppiness in the parameter space. The Bayesian formulation naturally regularizes the inverse problem and provides meaningful probabilistic constraints on otherwise underdetermined parameters.

To address the significant computational cost associated with repeated evaluations of the ET₃ forward operator, surrogate modelling techniques were introduced in Section 4. Reduced-order models, Gaussian process emulators, polynomial chaos expansions, and multi-fidelity correction strategies were employed to accelerate inference while maintaining fidelity to the underlying physics. These methods enable efficient sampling of high-dimensional posterior distributions using Markov Chain Monte Carlo and related algorithms.

The numerical and methodological developments presented in this work establish a scalable framework for uncertainty quantification in complex reactive nonequilibrium systems. The posterior distributions obtained not only provide point estimates of transport coefficients but also quantify epistemic uncertainty and parameter correlations, offering deeper insight into the structure of ET₃ closures. Moreover, uncertainty propagation through the forward model enables rigorous prediction intervals for macroscopic observables, including temperature, pressure, fluxes, and higher-order moment fields.

Despite these advances, several important directions remain open for future research. First, the calibration of ET₃ transport coefficients should be extended to incorporate experimentally derived datasets from high-energy-density physics, shock-tube experiments, and plasma diagnostics, enabling validation of the proposed framework against real physical systems. Second, the incorporation of more sophisticated model discrepancy representations, including hierarchical Bayesian models and stochastic partial differential equation corrections, would improve robustness in the presence of structural modelling errors.

Third, further development is required in the direction of scalable inference algorithms. While Gaussian process surrogates and reduced-order models significantly reduce computational cost, high-dimensional parameter regimes typical of chemically reacting multicomponent systems may benefit from advanced techniques such as normalizing flows, diffusion-based generative models, and physics-informed neural operators. These approaches may offer improved scalability and better handling of highly nonlinear posterior geometries.

Finally, the integration of the present Bayesian ET₃ framework with data-driven discovery methods presents a promising avenue for future work. In particular, hybrid approaches combining kinetic theory, Extended Thermodynamics, and machine learning may enable the automated discovery of closure relations and transport laws consistent with entropy principles and relativistic invariance.

In summary, this work establishes a consistent probabilistic framework for the identification and uncertainty quantification of third-order transport processes in reactive nonequilibrium flows. By combining rigorous thermodynamic modelling with modern Bayesian inference and surrogate acceleration strategies, the proposed approach provides a foundation for reliable, data-informed prediction of complex relativistic and chemically reacting systems. References