PaperlayerView exact revision on arXiv

Source-linked AI summary

Scalable Bayesian Optimization of Composite Functions for Image-Based Inverse Problems in Materials Characterization

Dasol Yoon, Poompol Buathong, Chia-Hao Lee, Yujia Zhang, David A. Muller, Peter I. Frazier

arXiv:2609.02126v1cs.LGcond-mat.mtrl-sci

TL;DR

Estimating specimen thickness and mistilt from PACBED images requires costly physics-based simulations, while existing grid-search and neural-network approaches have scaling or transferability limitations. SBOCF exploits the image objective’s composite structure through patch-level summaries and correction terms while preserving the pixel-wise SSE. Across synthetic and experimental SrTiO3 benchmarks, it improved optimization and parameter estimation without task-specific pretraining.

  • Problem

    PACBED-based estimation of specimen thickness and mistilt is an expensive simulation-matching inverse problem, and existing grid searches or neural networks have scaling or transferability limitations.

  • Method

    SBOCF replaces pixel-level intermediate outputs with patch-level representations and correction terms while retaining the original pixel-wise SSE objective.

  • Results

    Across synthetic and experimental SrTiO3 benchmarks, SBOCF achieved the strongest overall performance, including 290× final-SSE improvement over BO(EI) for Sim380 and 1.08× for Exp380.

  • Takeaways & Limitations

    SBOCF supports simulation-efficient PACBED parameter estimation without exhaustive search or task-specific pretraining.

  • Takeaways & Limitations

    Direct pixel-level BOCF is computationally impractical for typical PACBED images, and patch aggregation discards spatial information so its minimizers generally differ from those of the original objective.

Abstract

from arXiv · show

Estimating physical parameters from scientific images is a common inverse problem in materials characterization that often relies on expensive physics-based simulations. In electron microscopy, specimen thickness and crystal mistilt are critical parameters that govern how electrons scatter through the sample, and therefore the accuracy of any atomic-scale structure recovered from it. They are commonly inferred by matching experimental position-averaged convergent-beam electron diffraction (PACBED) patterns to simulated ones, but grid searches scale poorly and neural-network methods require extensive pretraining that may not transfer to new conditions. Here, we propose scalable Bayesian optimization of composite functions (SBOCF), a simulation-efficient method that exploits the known composite structure of the image-matching objective and the intermediate information contained in simulated images. By representing PACBED images with patch-level summaries and two correction terms, SBOCF preserves the original pixel-wise objective while reducing the number of modeled outputs from 24,649 to 11. Under a budget of 50 simulator evaluations, SBOCF outperformed standard Bayesian optimization with expected improvement on synthetic SrTiO3 benchmarks with thick and thin specimens, reducing the median final SSE by up to 290x in the thick-sample case. On experimental data, SBOCF produced parameter estimates consistent with previously reported values without task-specific pretraining. For a simulated mistilted specimen, using the SBOCF estimates in a downstream ptychographic reconstruction recovered sharp atoms that were otherwise blurred. These results establish SBOCF as a promising approach for inverse problems involving expensive simulators and high-dimensional structured outputs.

1 Introduction

PACBED-based estimation addresses the costly inverse problem of recovering specimen thickness and crystal mistilt from electron-diffraction images. SBOCF uses Bayesian optimization and a lower-dimensional composite representation to reduce simulation demands while retaining image-matching information.

  • Motivation: Physics-based simulation matching is a costly inverse-problem strategy for estimating unknown material parameters.The paper presents electron microscopy as a representative case where simulation cost can limit characterization throughput.
  • Motivation: Thickness and crystal mistilt affect electron scattering and are needed for accurate atomic-structure reconstruction.Thickness controls how many times electrons scatter, while mistilt is the angular deviation from the crystallographic zone axis.
  • PACBED estimation: PACBED patterns provide a measurement-based route to estimating thickness and mistilt through their thickness-sensitive fringe structure and tilt-sensitive symmetry.PACBED is formed by averaging diffraction patterns acquired across probe positions spanning at least one unit cell.
  • Existing approaches: Grid searches exhaustively simulate dense parameter grids, whereas neural-network approaches require large simulated libraries and retraining for new materials or imaging conditions.These approaches therefore trade computational cost against transferability.
  • Proposed approach: SBOCF partitions images into patches and models correction terms, reducing modeled outputs by more than 2,000-fold while exploiting the composite objective structure.The method is designed to avoid exhaustive search and task-specific training data.
  • Workflow: The workflow estimates specimen parameters before reconstruction by comparing experimental PACBED patterns with multislice simulations and adaptively selecting candidates.This separates parameter estimation from the subsequent reconstruction workflow.

2 PACBED Parameter Estimation with Bayesian Optimization

The paper formulates PACBED parameter estimation as expensive black-box optimization over simulated images and explains how Bayesian optimization exploits their composite structure. Standard BOCF is computationally impractical at PACBED resolution, motivating a lower-dimensional alternative.

  • Problem formulation: The problem estimates three parameters: specimen thickness and beam tilts along the x- and y-axes.The parameter vector lies in a three-dimensional specimen-parameter space.
  • Problem formulation: PACBED patterns vary non-monotonically with thickness and break symmetry direction-dependently under orthogonal tilts.These visual changes provide structure for distinguishing thickness and tilt effects.
  • Problem formulation: A multislice simulator maps candidate parameters to PACBED images, whose discrepancy is measured with pixel-wise sum of squared errors.The image-matching objective compares simulated and experimental PACBED intensities pixel by pixel.
  • Bayesian optimization: Bayesian optimization sequentially selects simulation candidates using a surrogate model that balances promising objective values against posterior uncertainty.Expected improvement is the primary acquisition function, with knowledge gradient considered as an additional baseline.
  • Composite objectives: The PACBED objective has composite structure because parameters first produce an image and pixel intensities are then combined by the SSE function.This intermediate-image structure can be exploited rather than modeling only the scalar objective.
  • Composite objectives: BOCF models intermediate pixel intensities with a multi-output Gaussian process and propagates uncertainty through the known outer SSE function.Because the resulting objective distribution is generally non-Gaussian, expected improvement is estimated by Monte Carlo sampling.
  • Limitations of standard BOCF: Direct BOCF requires modeling M^2 pixel intensities and becomes computationally impractical for typical PACBED images with M ≳128.This limitation motivates replacing pixel-level outputs with a lower-dimensional intermediate representation.

3 Proposed Method: Scalable Bayesian Optimization of Composite Functions

SBOCF makes composite Bayesian optimization scalable for PACBED matching by replacing pixel-level outputs with patch summaries and correction terms while preserving the original pixel-wise SSE objective.

  • SBOCF replaces the pixel-level intermediate output with lower-dimensional patch-level representations while retaining the original pixel-wise SSE objective.
  • Patch-level representation: Patch summaries aggregate PACBED pixels within disjoint patches, reducing the modeled intermediate quantities from M^2 pixel intensities to P summaries.
  • Patch-level representation: Aggregation discards within-patch spatial information, so the patch-level SSE minimizers generally differ from those of the original pixel-wise objective.
  • Correction for aggregation error: SBOCF represents the original objective as an input-dependent multiplicative correction of patch-level SSE plus a residual discrepancy.
  • Composite surrogate model: Independent GP surrogates model P + 2 reduced outputs, and the known outer function recovers the original pixel-wise SSE for expected-improvement optimization.
  • Sequential SBOCF procedure: The sequential procedure fits the reduced surrogates, propagates posterior samples through the outer function, selects the next candidate by maximizing EI, and repeats until the budget is exhausted.

4 Numerical Experiments

The numerical experiments compare SBOCF with Bayesian-optimization and random-sampling baselines on simulated and experimental SrTiO3 PACBED benchmarks under a fixed evaluation protocol.

  • Experimental design: The study compares SBOCF with BO(EI), BO(KG), and Random using a shared 7-point Sobol initialization and 50 total evaluations.
  • Evaluation metrics: Performance is measured by the best pixel-wise SSE at each iteration, alongside median SSE gain, parameter-estimation error, and acquisition-optimization time.
  • Simulated SrTiO3 PACBED: Synthetic SrTiO3 benchmarks use targets generated by the optimization simulator, allowing evaluation against known ground-truth specimen parameters.
  • Simulated SrTiO3 PACBED: The simulated benchmarks vary specimen thickness and beam tilts, including thick Sim380, thin Sim100, and Sim200 cases.
  • Additional analyses: Noise robustness and an alternative circular-center partition are evaluated in additional experiments.
  • Experimental SrTiO3 PACBED: Experimental Exp380 PACBED cannot be reproduced exactly by the simulator, so estimates are evaluated against reference thickness and tilt values reported in prior work.
  • Downstream MEP reconstruction: For downstream reconstruction, SBOCF-estimated thickness and mistilt from Sim200 are supplied to the corresponding 4D-STEM reconstruction.

5 Results and discussion

Across synthetic and experimental PACBED benchmarks, SBOCF achieved the strongest parameter-estimation performance, with especially large synthetic SSE gains and useful downstream reconstruction improvements.

  • Simulated SrTiO3 PACBED: SBOCF achieved the lowest final SSE and the most accurate, stable parameter estimates on the Sim380 synthetic benchmark.Both standard BO baselines outperformed random sampling.
  • Simulated SrTiO3 PACBED: 290× and 377× were the Sim380 final-SSE improvement factors over BO(EI) and BO(KG), respectively.Parameter-error improvement factors over BO(EI) were 24×, 26×, and 35× for thickness, tilt-X, and tilt-Y; over BO(KG), they were 23×, 22×, and 51×.
  • Simulated SrTiO3 PACBED: 47× and 320× were the Sim100 final-SSE improvement factors over BO(EI) and BO(KG), respectively.The corresponding parameter-error improvements were (1.9×, 9.4×, 14×) and (5.4×, 26×, 33×).
  • Runtime and accuracy: SBOCF had higher per-iteration cost than BO(EI) because it modeled 11 GP outputs, but achieved substantially lower final SSE and a favorable runtime–accuracy trade-off.It was also faster and more accurate than BO(KG), whose acquisition optimization was more computationally costly.
  • Experimental SrTiO3 PACBED: 1.08× and 1.13× were the experimental final-SSE improvement factors over BO(EI) and BO(KG), respectively, with SBOCF improving most parameter estimates.Measurement noise and simulation–experiment mismatch made these gains less pronounced than in synthetic cases.
  • Experimental SrTiO3 PACBED: 2.4× and 3.0× were the experimental thickness improvement factors relative to BO(EI) and BO(KG), respectively.The resulting estimates were close to the reference thickness of 380 Å and the tilt values reported in [6].
  • Experimental SrTiO3 PACBED: SBOCF produced competitive experimental parameter estimates without problem-specific neural-network training or a large simulated training set.It incurred greater acquisition-function overhead than BO(EI) but lower overhead than BO(KG).
  • Downstream MEP reconstruction: Using SBOCF estimates recovered sharp, vertical atomic columns in the simulated mistilted specimen, unlike reconstruction without tilt correction.The estimates matched ground truth within 5 Å in thickness and 0.1 mrad in tilt.

6 Conclusion

The paper introduces SBOCF for PACBED-based estimation of specimen thickness and mistilt. It reduces modeled outputs while preserving the pixel-wise objective, improves synthetic optimization, and yields experimentally consistent estimates without a large training dataset.

  • Conclusion: SBOCF estimates specimen thickness and mistilt from PACBED images using patch-level outputs and two correction terms.This preserves the pixel-wise SSE objective while reducing modeled outputs by approximately 2,000-fold relative to standard BOCF.
  • Conclusion: Under a limited simulation budget, SBOCF converged faster and achieved lower SSE than BO(EI), reducing median final SSE by approximately 290× on noise-free Sim380.On experimental data, its thickness and mistilt estimates were consistent with previous neural-network-based estimates without a large simulated training dataset.
  • Conclusion: Future work will extend SBOCF to additional PACBED-sensitive parameters, including probe aberrations and atomic-vibration amplitudes.The broader conclusion positions SBOCF as a practical approach for expensive-simulation inverse problems with high-dimensional structured outputs.

A Gaussian process posterior distribution

The Gaussian-process framework uses evaluated inputs and observations to form posterior predictions, which acquisition functions use to balance exploitation and exploration. The knowledge gradient targets expected improvement in the best inferred SSE, while a constrained decomposition derives closed-form choices for auxiliary variables.

  • The posterior uses evaluated inputs, a kernel matrix, prior mean, and observations to characterize predictions at a queried input.The kernel vector links the queried point to evaluated points, while the prior mean and covariance specify the GP prior.
  • Knowledge gradient measures the expected reduction in the best inferred SSE from an additional evaluation.Its expression combines the current best inferred value with the expected future best inferred SSE.
  • Knowledge gradient lacks an analytical formula and is evaluated using Monte Carlo approximation with one-shot optimization.
  • When f^P(x)>0, the constrained optimum projects f(x)/f^P(x) onto [10^-c, 10^c], then recovers δ(x) from the equality constraint.
  • When f^P(x)=0, δ(x) is fixed and any ϵ(x) in [10^-c, 10^c] is optimal, with c>0 user-specified.

E Optimization details

The optimization implementations use BoTorch with method-specific acquisition procedures and common Gaussian-process settings. SBOCF relies on Monte Carlo sampling and more acquisition-optimization restarts than the baseline methods.

  • All Bayesian optimization methods are implemented in BoTorch.
  • For EI, logEI is optimized with 20 restarts and 100 raw samples using BoTorch’s default multi-start procedure.The logEI formulation is used for greater numerical stability than standard EI.
  • For KG, the one-shot implementation uses 16 fantasy samples, reparameterization, automatic differentiation, 20 restarts, and 100 raw samples.
  • For SBOCF, the acquisition function uses 512 Sobol QMC samples with reparameterization and automatic differentiation, optimized using 50 restarts and 100 raw samples.

F Figure for Sim100

Figure F1 reports Sim100 results for a 100 Å specimen with tilts of (1.5, −1.5) mrad, using the same layout and conventions as Fig. 4.

  • Figure F1 shows the Sim100 results referenced in Section 5, with corresponding improvement factors reported there.
  • Sim100 uses ground-truth thickness 100 Å and tilts (1.5, −1.5) mrad.

G Figure for Sim200

Figure G1 reports Sim200 results for a 200 Å specimen with tilts of (3, −5) mrad. In this representative downstream-ptychography case, SBOCF estimates recover the ground truth closely.

  • SBOCF recovers Sim200 ground-truth parameters within 5 Å in thickness and 0.1 mrad in tilt.Sim200 is used as a representative case for the downstream ptychography application.
  • Sim200 uses ground-truth thickness 200 Å and tilts (3, −5) mrad.

H Ablation study: Simulated test cases with ground-truth noise

The ablation evaluates SBOCF under increasing Poisson noise across simulated PACBED problems. SBOCF remains competitive, but its advantage narrows with noise and parameter recovery degrades more strongly for thinner samples.

  • Noise settings: Peak photon counts of 150, 100, and 50 generate progressively higher Poisson noise in the simulated PACBED observations.The noisy benchmarks include Sim380, Sim100, and Sim200 variants at each specified photon-count level.
  • Evaluation metrics: Each noisy-case figure reports best observed pixel-wise SSE, final thickness and tilt estimates, and acquisition runtime versus final SSE.Panels (b)–(d) include dashed ground-truth parameter values for comparison.
  • Overall performance: SBOCF remains competitive across all noise levels and continues to provide accurate parameter estimates in many settings.The performance gap with standard BO narrows substantially under the highest noise setting, especially for thinner Sim100 problems.
  • Noise sensitivity: Under the highest noise, the performance gap between SBOCF and standard BO narrows substantially, particularly for thinner-sample Sim100 problems.Heavy pixel-level noise partially obscures the composite structure exploited by SBOCF.
  • Sample thickness: Thinner-sample cases are more sensitive to noise, whereas Sim380 retains clearer optimization improvements and more consistent parameter estimates.For Sim100 and Sim200, optimization quality and parameter consistency degrade more noticeably as noise increases.

I Ablation study: Alternative partition pattern

This ablation compares SBOCF’s regular square partition with an intensity-based domain partition across synthetic problems and noise levels. The square pattern generally matches or outperforms the domain pattern, especially for the thicker Sim380 case.

  • Partition designs: The domain pattern divides PACBED images into one central circular high-intensity region and four outer low-intensity quadrant regions.Unlike regular spatial patches, it separates broad intensity-based regions.
  • Experimental scope: The comparison covers Sim380, Sim100, and Sim200 under noiseless conditions and peak photon counts P150, P100, and P50.Figures report pixel-wise SSE and final thickness, tilt-X, and tilt-Y estimates.
  • Overall comparison: The square pattern performs better than, or at least comparably to, the domain pattern across the tested settings.This overall comparison includes both noiseless and noisy synthetic cases.
  • Thick samples: For Sim380, the square pattern reduces pixel-SSE faster and yields more stable final parameter estimates under noiseless and noisy observations.The corresponding comparisons span the Sim380 figures for noiseless data and peak photon counts P150, P100, and P50.
  • Thin samples: For thinner Sim100 and Sim200 problems, differences are smaller in some settings, but the square pattern generally gives comparable or lower final pixel-SSE and more stable estimates.The result holds across the tested partition comparisons, including noisy cases.
  • Interpretation: The domain pattern may be too coarse because broad intensity grouping can discard spatial information useful for distinguishing thickness and tilt effects.The square pattern preserves more local spatial information through smaller regular patches.
Loading 2609.02126v1…