Probabilistic Digital Twin in Foundation Construction

• Project foundation construction is a systematic approach to creating and optimizing geotechnical bases using data-driven, probabilistic digital twin frameworks and Bayesian inference.
• The framework employs real-time sensor integration, Monte Carlo simulation, and sequential Bayesian updating to quantify and manage uncertainties in soil behavior.
• Case studies reveal that using this approach can reduce costs and schedule overruns while ensuring safety through adaptive, risk-aware construction decisions.

Project foundation construction refers to the lifecycle of creating, managing, and optimizing the ground engineering base for civil infrastructure, with a technical emphasis on systematic uncertainty quantification, data integration, model updating, and decision optimization. The current paradigm, exemplified by the Probabilistic Digital Twin (PDT) framework, replaces traditional deterministic design with a rigorously probabilistic, data-driven, and continuously updating approach tailored for the geotechnical sector, accommodating aleatoric, data, model, and predictive uncertainties through Bayesian inference and stochastic optimization (Cotoarbă et al., 2024).

1. Conceptual Model: The Probabilistic Digital Twin in Foundation Construction

The PDT framework in foundation construction comprises three interdependent components:

• Physical State (PT): The actual ground–structure system, including soil, structure, and environmental context.
• Digital State (DT): A joint probabilistic distribution over all relevant model parameters and system states, encoding uncertainty.
• Bidirectional Data Streams:
  • Property data ($Z^{prop}$): Direct material measurements (e.g., lab, CPT).
  • Behavior data ($Z^{beh}$): In situ and structural performance readings (e.g., settlement, pore pressure).

The core loop operates in a cyclic manner—sense, infer, predict, act—collecting new data, updating digital state with Bayesian inference, generating probabilistic predictions of quantities of interest ($Q$), and optimizing construction decisions under ongoing uncertainty. Site learning is achieved by recursive probabilistic graphical model updates, maintaining the posterior belief $d_t = p(X_t \mid Z_{0:t}, U_{0:t-1})$ for the hidden physical state $X_t$.

2. Mathematical Framework for Uncertainty Quantification

Joint uncertainty is represented by the distribution

$$p(\boldsymbol\theta) = p(\sigma'_c, \gamma, E, c_v, c_h, \dots),$$

parameterized from empirical data (borehole, CPT) and expert judgment. Uncertainty propagation through forward models (e.g., settlement, bearing capacity) is performed by Monte Carlo integration:

$$p(y) = \int p(y | \boldsymbol\theta) p(\boldsymbol\theta) d\boldsymbol\theta \approx \frac{1}{N} \sum_{i=1}^{N} \delta(y - f(\boldsymbol\theta^{(i)})).$$

Bayesian updating with new data $D$ (e.g., CPT tip or settlement measure) applies:

$$p(\boldsymbol\theta \mid D) \propto p(D | \boldsymbol\theta) p(\boldsymbol\theta).$$

Temporal evolution and decomposition of total uncertainty arise via:

$$d_t(X_t) \propto \int d_{t-1}(X_{t-1})\, p(X_t \mid X_{t-1}, a_{t-1})\, p(Z_t \mid X_t, Q_t, a_{t-1})\, p(Q_t \mid X_t)\, dX_{t-1} dQ_t,$$

with $p(X_t \mid X_{t-1}, a_{t-1})$ as the action-conditioned state transition, $p(Z_t \mid X_t, Q_t, a_{t-1})$ as likelihood for observation, and $p(Q_t \mid X_t)$ representing the (possibly surrogate) behavior model.

3. Data Integration Workflow and Real-Time Model Updating

The operational pipeline encompasses:

1. Initialization: Assemble pre-construction datasets, including borehole logs and laboratory assays; construct initial Gaussian process–based probabilistic models of subsoil properties; define initial design action space.
2. Digital State Learning: Transform measurements via empirical correlations (with transformation/model uncertainty), interpolate with geostatistics (Kriging/GPR), and encode model-form uncertainty as covariance scaling or mixture distributions.
3. Behavioral Prediction Modeling: Select either physics-based or hybrid models for responses such as consolidation settlement ($S(t;\boldsymbol\theta)$), bearing capacity, or lateral pile deflections; surrogate models (e.g., polynomial chaos expansions, Gaussian Process, or neural networks) substitute for expensive computations, with explicit surrogate uncertainty quantification.
4. Online Monitoring and Bayesian Updating: Deploy field sensors (settlement plates, piezometers, inclinometers). At each timestep $t$, gather new behavior/property data, compute likelihoods, and update the state distribution using particle filters or Ensemble Kalman Filters.
5. Decision Optimization: Compute posterior predictive distributions of $Q_{t_{max}}$ for critical events. Expected costs or rewards are evaluated as

$$\mathbb{E}[C(\boldsymbol\theta, a_{0:t})] = \int C(\boldsymbol\theta, a_{0:t})\, p(\boldsymbol\theta \mid Z_{0:t}) d\boldsymbol\theta.$$

Control actions $a_t$ (e.g., surcharge increase, schedule shift) are optimized via POMDP formalism or heuristic stochastic optimization.

1. Implementation and Looping: Implement selected control actions, then iterate the entire cycle until project completion.

4. Case Study: Highway Embankment Foundation in Stockholm

A 550 m highway embankment on soft clay was used to demonstrate the PDT. Key aspects:

• Inputs: 9 borehole data series (e.g., $\sigma'_c$, $M_0$, unit weights), CPT-based permeability correlations, and weekly settlement monitoring.
• Modeling: Initial soil prior as lognormal distributions; consolidation modeled with Terzaghi–Hansbo theory. Online Bayesian updating performed with a particle filter for $p(S(t_{max}))$.
• Decision Policies: Replaced deterministic, uniform surcharging with an adaptive, data-driven strategy. Surcharge is increased only if posterior risk of missing $s_{target}$ exceeds a predefined tolerance; settlement is continuously monitored until the coefficient of variation subsides below target.

Quantitative results:

Strategy	Exp. Cost (M SEK)	Cost Std. (M SEK)	Cost Reduction
Deterministic (baseline)	8.11	7.40	—
PDT ($\sigma_\varepsilon=0.05$ m)	6.42	5.29	–21% (mean), –28% (std)
PDT ($\sigma_\varepsilon=0.15$ m)	7.60	4.43	6–13% (mean), up to 40% (std)

Safety constraints ($S(t_{max}) \leq s_{target},\ OCR \geq 1.10$) were consistently maintained with high confidence, yielding reduced schedule overruns and penalties (Cotoarbă et al., 2024).

5. Software Infrastructure and Computational Tools

Best practices recommend leveraging the Python ecosystem for rapid prototyping and scalability: NumPy/SciPy for statistics, PyMC3 or Stan for Bayesian inference, Scikit-learn or GPyTorch for Gaussian processes, TensorFlow/PyTorch for surrogate modeling, and additional libraries (e.g., filterpy, pykalman) for sequential Monte Carlo. Accelerated updates can be performed via HPC or GPU-enabled pipelines. Model-form uncertainty is addressed using ensemble surrogates or multimodel mixtures, and CPT–soil correlations are explicitly modeled for transformation errors.

Visualization of posterior evolutions and scenario analyses are essential for transparency and communication with stakeholders, who may use the PDT platform as a semi-automated extension of classical Eurocode 7 observational practices.

6. Stakeholder Collaboration and Lifecycle Integration

Effective deployment recommends:

• Early identification of project-specific risk tolerances and key output metrics.
• Clearly defined data collection protocols and anticipated data frequency/quality.
• Scalable design: initial simple (1D) models, upgradeable to multi-dimensional and FE models as data and compute become available.
• Tooling for reporting “What-If” scenarios to demonstrate the risk-reduction value of added monitoring.
• PDT architectures that support continuing model refinement and lifetime data integration, including operational-phase performance tracking.

This suggests that a well-structured, stakeholder-inclusive PDT process improves both technical and project outcomes throughout construction and operational phases (Cotoarbă et al., 2024).

7. Impact, Prospective Directions, and Recommendations

Adoption of the Probabilistic Digital Twin in geotechnical foundation construction yields statistically significant improvements in cost, risk, and schedule control—shown by 10–20% cost savings, substantially reduced cost variance, and consistently high safety confidence. The PDT approach generalizes beyond case studies to serve as a practical and scalable route to “smart” geotechnical design. Explicit modeling of all uncertainty sources, combined with real-time Bayesian updating and MCMC/SMC optimization, establishes a new standard for risk-aware ground engineering (Cotoarbă et al., 2024).

• Recommended practices: Systematic requirements analysis, modular software ecosystems, explicit uncertainty quantification at every step, and participatory governance.
• Anticipated advances: Increasingly automated, continuous-feedback twin systems with richer sensor arrays, real-time dashboards for all actors, and a broader embrace of integrated, probabilistic management throughout the civil infrastructure sector.