Probabilistic Digital Twin

Updated 13 January 2026

Probabilistic digital twins are computational models that replace deterministic logic with Bayesian, uncertainty-aware representations of physical systems.
They integrate physics-based simulations with data-driven surrogates to quantify aleatoric, epistemic, and model-form uncertainties for improved system reliability.
Advanced methods like sequential Bayesian filtering and multi-fidelity modeling enable real-time calibration and risk-aware decision support.

A probabilistic digital twin is a mathematical and computational construct that replaces the deterministic state/update logic of traditional digital twins with a fully probabilistic, uncertainty-aware representation of physical assets and their environment. This paradigm fuses physics-based modeling, modern probabilistic machine learning, and Bayesian inference, enabling the explicit representation, propagation, and calibration of aleatoric and epistemic uncertainties throughout simulation, monitoring, diagnostics, and decision-making tasks. Probabilistic digital twins are architected to address the inherent variability of complex real-world systems, improve trust and generalization, and inform risk-aware prediction, control, and maintenance actions.

1. Mathematical Foundations and Uncertainty Formulations

Probabilistic digital twins are formally grounded in Bayesian probability theory and the graphical-model (probabilistic graphical model, PGM) or stochastic process perspective. The asset-twin system is typically encoded as a coupled dynamical system, where the physical state, digital state, inputs, and measurements are all modeled as random variables evolving over time. The joint probabilistic model takes the general form:

$p(x_{0:T}, y_{1:T}, u_{1:T}; \theta) = p(\theta) p(x_0 \mid \theta) \prod_{t=1}^T p(x_t \mid x_{t-1}, u_t, \theta) p(y_t \mid x_t, \theta)$

where $x_t$ denotes the digital twin's latent state at time $t$ , $y_t$ the multivariate observation (possibly sensor data or high-dimensional signals), $u_t$ the exogenous input or control, and $\theta$ static or slow-varying parameters. This joint model is realized as a dynamic Bayesian network (DBN), hidden Markov model (HMM), or partially observable Markov decision process (POMDP), and may involve both discrete and continuous latent variables (Kapteyn et al., 2020, Torzoni et al., 2023, Varetti et al., 15 Dec 2025).

Uncertainties are categorized into:

Aleatoric: Inherent randomness in physical processes, captured by process and measurement noise, or random channel effects.
Epistemic: Parametric uncertainty from limited knowledge, e.g., uncertain calibration, unmodeled dynamics, or data scarcity.
Model-form: Uncertainty in model structure or hierarchical model choices, including surrogate-fidelity gaps or physics misspecification (Kashyap et al., 27 Nov 2025, Cotoarbă et al., 2024).
Data: Measurement or observation errors, modeled via explicit likelihoods with noise covariances.
Prediction: Uncertainty in forecasts of future quantities of interest (QoIs), propagated from upstream uncertainties (Cotoarbă et al., 2024).

Probabilistic digital twins update beliefs about system states and parameters via sequential Bayesian inference as new data arrives, enabling the continuous calibration of the digital state to the physical asset (Kapteyn et al., 2020, Torzoni et al., 2023).

2. Physics-Based, Data-Driven, and Hybrid Uncertainty Modeling

A central feature is the probabilistic hybridization of physics-based simulation with data-driven (machine learning or surrogate) models:

Physics-based surrogates are constructed by sampling or emulating high-fidelity physical models, then introducing randomization over parameter fields, model outputs, or unobservable states. For example, multiple independent calibrations of a power-line model produce an ensemble of digital twin instances whose empirical distribution quantifies parametric uncertainty (Das et al., 2023).
Surrogates (multifidelity, reduced-order, neural operator, or regression-based) are learned from simulation or observation data, augmented with Bayesian or Gaussian process corrections to account for residual error and local variance (Desai et al., 2023, Li et al., 2023, Liu et al., 2024). Multifidelity approaches cascade low- and high-fidelity models with probabilistic auto-regression, capturing the uncertainty across scales.
Machine learning modules (e.g., Bayesian neural networks, deep diffusions, DNN classifiers with stochastic or heteroscedastic heads) are embedded within the digital twin and trained to output calibrated predictive distributions, not point values (Kashyap et al., 27 Nov 2025, Das et al., 2023, Belousov et al., 2023).
Explicit uncertainty propagation is achieved via techniques such as assumed density filtering (ADF), which propagates input uncertainty through all neural network layers by matching moments for linear/affine and nonlinear (e.g., ReLU) transformations (Das et al., 2023).

3. Bayesian Inference and Online Sequential Updating

Bayesian filtering (Kalman, particle, or sequential Monte Carlo) constitutes the backbone of probabilistic digital twin updating. At each time step, observed data incrementally refines the posterior over system states and parameters:

$p(x_t \mid y_{1:t}, u_{1:t}) \propto p(y_t \mid x_t, \theta) \int p(x_t \mid x_{t-1}, u_t, \theta) p(x_{t-1} \mid y_{1:t-1}, u_{1:t-1}) dx_{t-1}$

High-dimensional or hybrid discrete-continuous models employ sum-product or message-passing algorithms (for PGMs), extended/unscented Kalman filtering for quasi-linear-Gaussian systems (Agarwal et al., 1 Nov 2025), or particle filtering for strongly nonlinear, high-variability systems (Cotoarbă et al., 2024, Desai et al., 2023).

Parameter posteriors, e.g., for transition matrices, are updated online via conjugate Bayesian rules (Dirichlet-multinomial for discrete transitions, Gaussian for continuous parameters), achieving adaptive, data-driven transition models (Varetti et al., 15 Dec 2025, Tezzele et al., 2024). Risk-averse inference leverages distributional risk measures (VaR, CVaR) to bias decision-making policy updates towards robustness under rare events (Tezzele et al., 2024).

Process structure may also be periodically refit using sparse Bayesian learning or spike-and-slab priors for parameter selection, maintaining interpretability and sharp uncertainty quantification in model expressions (Tripura et al., 2022).

4. Quantification, Decomposition, and Exploitation of Predictive Uncertainty

Total predictive uncertainty is systematically decomposed:

$\widehat\sigma^2_\mathrm{total} = \widehat\sigma^2_\mathrm{al} + \widehat\sigma^2_\mathrm{ep}$

where $\widehat\sigma^2_\mathrm{al}$ (aleatoric variance) is estimated via explicit moment propagation (e.g., in ADF) or learned as the output variance/multivariate Gaussian head, and $\widehat\sigma^2_\mathrm{ep}$ (epistemic variance) is quantified by stochastic forward passes (e.g., Monte Carlo dropout) or Bayesian ensembles (Das et al., 2023, Belousov et al., 2023). This decomposition is essential for downstream decision-making: high total variance regions are flagged for human intervention, data acquisition, or conservative actions.

In generative settings, fully probabilistic models like conditional denoising diffusion probabilistic models (DDPMs) realize an expressive stochastic mapping from latent variables or digital templates to physical measurements, precisely matching the distributional properties (variability, higher-order moments) observed in real production processes (Belousov et al., 2023).

5. Decision-Making and Control Under Uncertainty

Probabilistic digital twins enable risk-aware, optimal, and adaptive decision support by embedding the entire digital twin state into Markov decision processes (MDPs), parametric MDPs, or partially observable MDPs (POMDPs) with explicit dependence on the updated beliefs (Agrell et al., 2021, Torzoni et al., 2023, Varetti et al., 15 Dec 2025).

Value iteration or policy optimization is performed over the current posterior, exploiting all uncertainty information. Probabilistic scheduling in construction integrates Bayesian duration-posterior sampling within Monte Carlo CPM rollouts; critical-path and buffer-risk indices are tracked throughout (Khoshkonesh et al., 4 Nov 2025). In maintenance and adaptive planning for mission-critical assets, risk measures such as CVaR drive the policy optimization step, ensuring a tail-robust operating policy that balances cost and rare-event safety (Tezzele et al., 2024).

Active inference approaches further unify probabilistic modeling, state estimation, and decision-making by optimizing information gain and goal-alignment through a variational free energy objective, enabling adaptive, epistemic-exploratory action selection (Torzoni et al., 17 Jun 2025).

6. Architecture, Surrogate Construction, and Practical Implementation

A typical probabilistic digital twin system integrates:

Physics-based simulation engines (for offline database generation or online core predictions), sometimes accelerated by reduced-order models (ROMs) or high-dimensional operator surrogates such as Fourier Neural Operators (FNOs) (Liu et al., 2024).
Data-driven surrogates: fitted from physics simulation and/or observation data, using:
- Multi-fidelity hierarchical surrogates (deep-H-PCFE) (Desai et al., 2023)
- Deep generative models or neural operators (Belousov et al., 2023, Liu et al., 2024)
Sensor data assimilation: via EKF, SMC, or Bayesian filters updating state and parameter posteriors in real time (Agarwal et al., 1 Nov 2025, Cotoarbă et al., 2024).
Uncertainty-aware monitoring and anomaly detection: leveraging state-posterior covariance and data likelihood residuals (Agarwal et al., 1 Nov 2025).

Practical design patterns include combining multi-instance physics-twin ensembles (for parametric and structural uncertainty), propagation and fusion of uncertainty throughout the surrogate and decision layers, and continuous online retraining or Bayesian updating as real data becomes available (Das et al., 2023, Desai et al., 2023, Varetti et al., 15 Dec 2025).

7. Applications, Performance, and Generalization

Probabilistic digital twin methodologies have demonstrated efficacy across diverse domains, including:

Structural health monitoring: Explicit tracking of system degradation and damage with Bayesian inference on health state, enabling timely maintenance and risk-aware operation (Torzoni et al., 2023, Kashyap et al., 27 Nov 2025, Varetti et al., 15 Dec 2025).
Additive manufacturing: Physics-informed ML surrogates infer stochastic melt-pool and defect formation, with Bayesian calibration against empirical measurements, supporting control and QA applications (Li et al., 2023, Liu et al., 2024).
Power-line and infrastructure monitoring: DNN surrogates trained on uncertainty-augmented synthetic data provide calibrated diagnostics under sparse or variable data (Das et al., 2023).
Nuclear transient diagnostics: Ensemble digital twin architectures combine specialized models with probabilistic voting for surrogate tracking and safety metric estimation (Chen et al., 2022).
Air traffic control simulation: Probabilistic digital twins of operational airspace combine PIML trajectory models with uncertainty-quantified scenario generation for AI agent training and human-in-the-loop evaluation (Pepper et al., 6 Jan 2026).
Geotechnical engineering and construction: Probabilistic workflows systematically propagate uncertainty in subsoil properties, model-form, and observations, outperforming heuristic or rule-based baselines in predictive accuracy and risk reduction (Cotoarbă et al., 2024).

Empirical results report substantial improvements in calibration, predictive interval sharpness, decision robustness, and cross-domain generalization compared to deterministic or heuristic digital twin baselines, provided that probabilistic surrogates and uncertainty-aware control policies are correctly implemented.

Probabilistic digital twins thus represent a unifying, rigorously Bayesian extension of the digital twin paradigm, delivering mathematically principled, uncertainty-aware monitoring, prediction, and decision support across cyber-physical domains. Their systematic treatment of uncertainty is essential for reliable diagnostics, robust optimization, and trustworthy autonomy in safety-critical and data-sparse environments.