Bayesian Co-Navigation Framework

• Bayesian co-navigation is a framework that concurrently navigates experimental and model parameter spaces using Gaussian process surrogates to minimize epistemic uncertainty.
• It employs a three-loop workflow—experimental, theoretical, and hyperparameter updates—to drive rapid, on-the-fly model calibration and adaptive digital twin construction.
• Applications span materials science digital twins and human–robot teleoperation, significantly reducing experimental queries and balancing cost with computational latency.

Bayesian co-navigation is a framework for real-time, recursive integration of experimental data and theoretical models in automated scientific or engineering workflows. The central principle is the concurrent navigation of both experiment and model parameter spaces, dynamically calibrating models with on-the-fly experimental feedback to minimize epistemic uncertainty. Bayesian co-navigation thereby enables the autonomous, adaptive construction of digital twins that are not only data-correlative but also physically interpretable, providing mechanistic insight while balancing the cost and latency of experimental and computational components (Slautin et al., 8 Dec 2025); (Slautin et al., 19 Apr 2024).

1. Mathematical Structure of Bayesian Co-Navigation

Bayesian co-navigation formalizes the coordination between experimental observables and physical model responses under uncertainty. The approach relies on the following stochastic processes:

• Experimental mapping $f_E(x)$, where $x$ indexes the object or condition space (e.g., material composition, domain pattern).
• Theoretical mapping $f_T(x;\theta)$ from the computational physical model, parameterized by hyperparameters $\theta$.
• Hyperparameter domain $\Theta$ with prior $p(\theta)$, and an auxiliary posterior $p(\theta|D)$ continually updated from data.

At each iteration, observed data

• $D_E = \{(x_i, y^{(E)}_i)\}$
• $D_T = \{(x_j, y^{(T)}_j)\}$

are incorporated via Gaussian process (GP) surrogates:

• $g_E(x) \approx f_E(x)$
• $g_T(x; \theta) \approx f_T(x; \theta)$

Posterior inference for $\theta$ uses the likelihood of experimental/model deviations, e.g.,

$$p(D|\theta) \propto \exp \left( -\frac{1}{2\sigma_y^2} \sum_i \left[ R^{exp}(x_i) - R^{sim}(x_i; \theta) \right]^2 \right)$$

and is explored via a GP surrogate over $\theta \to \operatorname{MSE}(\theta)$:

$$f(\theta) = \operatorname{MSE}(\theta) = \frac{1}{N}\sum_i [R^{exp}(x_i) - R^{sim}(x_i; \theta)]^2$$

Acquisition functions guide active data collection and model refinement:

• Experimental acquisition: select $x_{next} = \arg \max_x \sigma_{GP}^{exp}(x)$
• Theoretical update: select $$\theta_{next} = \arg\min_\theta \alpha_{LCB}(\theta)$$, with

$$\alpha_{LCB}(\theta) = \mu_{GP}(\theta) - \kappa \sigma_{GP}(\theta)$$

where $\kappa$ tunes exploration/exploitation.

The outer loop seeks $\theta^* = \arg\min_\theta U(\theta)$, $U(\theta)$ being an empirical estimate of model–experiment mismatch:

$$U(\theta) = \frac{1}{|\mathcal X'|} \sum_{x \in \mathcal X'} [g_T(x; \theta) - g_E(x)]^2$$

(Slautin et al., 8 Dec 2025); (Slautin et al., 19 Apr 2024).

2. Three-Loop Workflow Architecture

Bayesian co-navigation generally operates through three nested, partially asynchronous loops:

1. Experimental Loop (E-loop): Actively selects new experimental conditions $x$ based on current uncertainty or information gain. Conducts measurements and updates the corresponding GP surrogate.
2. Theoretical Loop (T-loop): For given $\theta$, performs new physical model simulations over a batch of $x$, updates the simulation GP.
3. Hyperparameter (Θ-loop): At regular intervals, evaluates the model–experiment epistemic gap and updates $\theta$ using Bayesian optimization over the GP surrogate.

An exemplary workflow (Slautin et al., 8 Dec 2025):

• Initialize with a batch of experimental and simulation data.
• Iteratively alternate: (a) run batches of simulations at proposed $\theta$; (b) perform new experiments at most uncertain $x$; (c) retrain surrogates; (d) Bayesian optimize $\theta$ to minimize epistemic gap.
• All data and surrogates are synchronized at each theory update, driving convergence.

Pseudocode and detailed maintenance of the three-loops structure is provided in (Slautin et al., 19 Apr 2024).

3. Surrogate Modeling and Acquisition Functions

Surrogate models are central, enabling cheap emulation of both experimental and theoretical response functions and facilitating data-efficient Bayesian optimization:

• Gaussian Process Surrogates: $g_E(x)$ and $g_T(x;\theta)$ model noisy, high-dimensional response landscapes. For high-dimensional $x$, deep kernel learning or feature embedding may be used.
• Acquisition in E-/T-loops: Maximum uncertainty ($\sigma$), upper confidence bound (UCB), expected improvement, or lower confidence bound (LCB) guide adaptive sampling.
• Hyperparameter Optimization: An independent surrogate $g_\theta(\theta)$ approximates the epistemic gap $U(\theta)$, supporting outer-loop Bayesian optimization to recalibrate the theoretical model.

This approach balances experimental and computational resources, handling asynchronous operation and rate disparities by adjusting loop cadence and acquisition criteria (Slautin et al., 19 Apr 2024).

4. Applications and Case Studies

Digital Twins in Materials Science

In materials combinatorics and digital twin construction, Bayesian co-navigation has enabled the tight integration of automated atomic force microscopy (AFM) with kMC models of thin-film systems (Slautin et al., 8 Dec 2025) and adaptive design of ferroelectric digital twins (Slautin et al., 19 Apr 2024). Key examples:

System	Experimental Loop	Theoretical Model	Result
(CrTaWV)$_x$-Mo$_{1-x}$ films	Automated AFM	kMC growth, 3 hyperparams	5$\times$ reduction in AFM scans, mechanistic insight into surface diffusion barriers
PbTiO$_3$ domain patterns	BEPS, PFM microscopy	FerroSim, Ginzburg-Landau	Order-of-magnitude reduction in digital twin error

Both studies demonstrate:

• Rapid convergence to accurate experimental predictions with far fewer experimental queries than exhaustive sampling.
• On-the-fly model refinement, enabling interpretable estimates of physical parameters (e.g., effective bond energies or coupling constants).
• The uncovering of mechanistic phenomena directly from co-navigation: e.g., the role of hetero-bonding in roughness maxima (Slautin et al., 8 Dec 2025).

Human–Robot Co-Navigation

Bayesian co-navigation also extends to adaptive human–robot teleoperation. In (Panagopoulos et al., 2021), Bayesian Operator Intent Recognition (BOIR) fuses observations and explicit actions to recursively infer human intent, enabling mixed-initiative shared control:

• Observes geometric cues (bearing, path length) and fuses them via weighted exponentials in the observation model.
• Action model (AIRM) incorporates asynchronous explicit intent.
• Posterior over discrete goal space $G$ is updated recursively, guiding adaptive robot assistance.

BOIR outperforms baselines in accuracy and log-loss, especially in complex or ambiguous environments (Panagopoulos et al., 2021).

5. Measurement and Minimization of Epistemic Uncertainty

A central metric is the epistemic gap—quantifying the disagreement between experimental and theoretical surrogates:

$$\Delta_i = \int [\mu_{GP}^{exp}(x) - \mu_{GP}^{sim}(x)]^2 dx$$

Minimization of $\Delta_i$ through outer-loop hyperparameter calibration ensures the digital twin remains aligned with experimental observations and theoretical principles. The Θ-loop's optimization problem is formally:

$\theta^* = \arg\min_\theta U(\theta)$

where $U(\theta)$ is estimated as described above. This continual gap minimization embodies the real-time self-correction of the digital twin (Slautin et al., 8 Dec 2025); (Slautin et al., 19 Apr 2024).

6. Practical Considerations and System Integration

• Hardware–Software Coupling: Realizations integrate autonomous laboratory equipment (e.g., AFM with real-time Python control) and high-throughput simulation on CPUs/GPUs (Slautin et al., 8 Dec 2025).
• Latency and Cost Balancing: The system adjusts E-/T-loop update rates to accommodate mismatched timescales (e.g., simulation per run $\sim$ seconds, experiment $\sim$ minutes) (Slautin et al., 19 Apr 2024).
• Parallelization: Experimental and theoretical loops can operate asynchronously, synchronizing only for global $\theta$ updates and surrogate retraining.
• Extension Potential: The framework is agnostic to the specific physical system, suitable for materials, molecular simulations, optical/nanocluster properties, or any scenario where model–experiment integration is required.

7. Limitations, Extensions, and Research Directions

While Bayesian co-navigation presently delivers significant reductions in experimental burden and rapid model interpretability, limitations include:

• Surrogate modeling scalability in extreme high-dimensional spaces.
• Hyperparameter tuning of acquisition function parameters ($\kappa$, $\beta$) and prior selection, often done heuristically (Panagopoulos et al., 2021).
• Fixed set of possible actions/goals in human–robot co-navigation (Panagopoulos et al., 2021).

Ongoing and future work includes:

• Data-driven tuning of acquisition and transition model parameters.
• Extension to adaptive goal discovery, richer observation streams (gaze, language, multisensory integration), and embedding BOIR within full POMDP planners (Panagopoulos et al., 2021).
• Generalization across scientific domains, enabling “theory-in-the-loop” closed-loop design of experiments for molecular, microstructural, and materials systems (Slautin et al., 19 Apr 2024).

Bayesian co-navigation thus operationalizes the dynamic feedback between theory and experiment for autonomous discovery and control, with broad applicability to materials design, scientific automation, and human–machine teaming.