Autonomous SPM Framework

• Autonomous SPM is an integrated system that automates experimental decision-making using Bayesian optimization and closed-loop feedback for efficient, high-resolution measurements.
• It leverages advanced acquisition functions (UCB, EI, PI) and memory-based sampling to reduce tip wear, minimize drift, and accelerate high-value region discovery.
• The framework is extensible with image-based priors and multi-objective strategies, enabling scalable, adaptive exploration for complex spectroscopic tasks.

Autonomous Scanning Probe Microscopy (SPM) frameworks constitute a class of integrated hardware/software systems designed to automate experimental decision-making and data acquisition in scanning probe microscopy modalities, such as atomic force microscopy (AFM), piezoresponse force microscopy (PFM), and scanning tunneling microscopy (STM). These frameworks leverage structured machine learning, predominantly Bayesian optimization (BO), to guide measurement location selection, optimize parameter spaces, and interpret high-dimensional spectroscopy data. In advanced implementations, autonomous SPM integrates prior physical knowledge, cost-aware planning, uncertainty quantification, and closed-loop feedback to achieve efficient, reproducible, and adaptive experimental workflows. Typical benefits include order-of-magnitude acceleration in high-value region discovery, reduced tip wear, increased sample stability, and the enablement of complex spectroscopic tasks that are otherwise infeasible under conventional raster or manual protocols (Vasudevan et al., 2020).

1. Bayesian Optimization Algorithms for Autonomous SPM

Bayesian optimization forms the core of autonomous SPM, facilitating adaptive sampling of the spatial domain to maximize or learn an objective function, commonly a scalarized physical property, such as the hysteresis-loop area in PFM electromechanical mapping. The framework models noisy measurements as $y_n = f(x_n) + \epsilon_n$, with $\epsilon_n \sim \mathcal{N}(0,\sigma_n^2)$ (Vasudevan et al., 2020). A Gaussian process (GP) surrogate is fitted over all previous measurements, yielding a predictive mean $\mu_n(x)$ and variance $\sigma_n^2(x)$ at each unsampled location $x$:

$$\mu_n(x) = k_n(x)^\top [K_n + \sigma_n^2 I]^{-1} y,\qquad \sigma_n^2(x) = k(x,x) - k_n(x)^\top [K_n + \sigma_n^2 I]^{-1} k_n(x)$$

Acquisition functions drive the selection of future probe points:

• Upper Confidence Bound (UCB): $\alpha_{UCB}(x) = a \mu_n(x) + b \sigma_n(x)$ (commonly $a=1$, $b=\kappa>0$)
• Expected Improvement (EI): $$\alpha_{EI}(x) = (\mu_n(x) - y^+ - \xi)\Phi(Z) + \sigma_n(x)\phi(Z)$$, with $Z = (\mu_n(x) - y^+ - \xi)/\sigma_n(x)$
• Probability of Improvement (PI): $$\alpha_{PI}(x) = \Phi( (\mu_n(x) - y^+ - \xi) / \sigma_n(x) )$$

where $y^+ = \max_{i\leq n} \mu_n(x_i)$, $\Phi$ and $\phi$ are the standard normal cumulative and probability density functions, and $\xi \geq 0$ determines an exploration-exploitation margin.

Path-finding and memory-penalty modifications, such as short-term memory spatial penalization (Vasudevan et al., 2020), further refine the acquisition landscape to minimize redundant sampling and efficiently traverse high-response clusters:

$$\tilde{\alpha}(x) = \alpha(x) - \sum_{t=1}^T \gamma^{t-1} / \|x - x_t\|$$

Batch acquisition selects the top-$B$ points after memory/distance filtering, allowing for hardware-efficient multitasking.

2. Experimental Workflow and Hardware Integration

Autonomous SPM workflows instantiate a multi-stage pipeline tightly coupled with microscope hardware (Vasudevan et al., 2020, Ziatdinov et al., 2022):

• Initialization: Random $N_0$-point seed grid. Each location is spectroscopically measured and scalarized (e.g., loop area extraction from PFM sweeps).
• BO/Active-Learning Loop:

1. Update GP posterior including all measured data. 2. Compute acquisition values for unsampled points. 3. Apply memory/distance filter and select next batch. 4. Transfer the batch to the control PC. 5. Hardware executes voltage waveforms, records response, computes scalar metric, and returns data.

• Termination: Pre-set sample budget or convergence in $\mu_n, \sigma_n$.

Integration is realized using platforms such as Asylum Research Cypher AFM interfaced by LabView/National Instruments DAQ. The GP/BO engine runs Python/CUDA code, often on high-performance compute infrastructure (e.g., NVIDIA DGX-2), and inter-process communication is maintained via TCP/UDP. Typical image acquisition and BO loop timing is sub-second to ~6 seconds per iteration. Real-time data processing and batch selection ensure adaptive control with minimized experiment latency.

3. Performance Benchmarks and Comparative Efficiency

Empirical validation demonstrates major improvements over manual and grid baseline protocols (Vasudevan et al., 2020):

Sampling Protocol	Points Acquired	Time (sec)	High-Value Pixel Capture	Speedup Factor
Raster Scan	2,500	5,875	100%	1×
Autonomous BO	400	1,180	>80%	~5×
Random Sampling	400	N/A	<50%	N/A

As reported, autonomous BO typically achieves coverage of high electromechanical response regions with 3–5× fewer measurements. Clustering and spatial exploration are substantially improved, with two out of three major clusters reliably identified within a 400-point budget. Simulations reinforce the ability to recover the top 10% high-response pixels within three-fold fewer samples compared to uniform scans.

Indirect metrics indicate reduced tip wear—due to minimized lateral motions and sparse sampling—and demonstrate increased sample stability, permitting spectroscopic sequences that are precluded by drift or damage in full-grid scanning.

4. Extensions: Prior Information and Advanced Acquisition Strategies

To exploit additional prior information, the framework allows:

• Image-based Priors: Fast single-frequency PFM images are incorporated as secondary channels to the GP input; features $\phi(x)$ extracted via convolutional neural networks (CNN) inform spatial kernels:

  $$k((x,\phi(x)),(x',\phi(x'))) = \sigma_f^2\,\exp(-\|x-x'\|^2/(2\ell^2) - \|\phi(x)-\phi(x')\|^2/(2\ell_\phi^2))$$

This deep-kernel approach generalizes the BO search to include domain boundaries and spatial covariance structure (Vasudevan et al., 2020).

• Custom/Entropy-Driven Acquisition: Acquisition functions are extended with sample-specific cost terms (e.g., time-to-move), entropy-based mutual information criteria, and multi-objective BO to simultaneously optimize multiple physical targets (e.g., loop area vs. scan time, sample drift) (Vasudevan et al., 2020).
• Batch and Parallel BO: Larger batch sizes are selected under pairwise diversity constraints (kernel length-scale), and sparse GP solvers or distributed schemes are proposed for scaling to larger domains.

5. Pseudocode Summary and Open-Source Implementation

A distilled pseudocode for the autonomous SPM loop is as follows (Vasudevan et al., 2020):

Initialize: X ← random N₀ seed positions Y ← measure_loop_area(X) For n = N₀ to N_max step B: (μ,σ) ← GP_train(X, Y; kernel=RBF, noise=σ_n²) α(x) ← acquisition(μ(x), σ(x)) # EI, PI, or CB α̃(x) ← α(x) – Σ_{t=1..T} γ^{t-1}/‖x–X_{n-t+1}‖ X_next ← top-B of α̃(x) (pairwise dist ≥ ℓ) Y_next ← measure_loop_area(X_next) Append (X, Y) ← (X ∪ X_next, Y ∪ Y_next) End

Implementation is available in the open-source GPim package. The integration of GPU-accelerated GP-BO with low-level experimental control enables dramatic acceleration of SPM-based discovery and provides a foundation for physics-informed, multi-objective, and high-dimensional autonomous microscopy.

6. Future Development: Toward Physics-Informed, Multi-Objective Autonomous SPM

Recent framework enhancements propose integration with structured hypothesis learning, DKL-based priors, and multi-objective BO to generalize beyond single-scalar objectives. Such developments are expected to catalyze progress in combinatorial materials exploration, destructive or irreversible measurement modalities, and complex structure–property landscape mapping. Prospects include automated experiment workflows tied to upstream synthesis, batch-mode acquisition for throughput scaling, and fully unattended "self-driving" SPM laboratories (Vasudevan et al., 2020, Ziatdinov et al., 2022).

Taken together, the autonomous SPM framework described represents a rigorous synthesis of Bayesian, computational, and hardware innovations that transform traditional probe microscopy into adaptive, efficient, and reproducible scientific instrumentation.