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arxiv: 2606.11240 · v2 · pith:ZFE4ZFCSnew · submitted 2026-05-29 · ⚛️ physics.comp-ph · cond-mat.str-el· cs.LG· quant-ph

Physically Constrained Ensemble Gaussian Process Modelling for Expensive Quantum Systems with Heteroskedastic Noise

Arpan Biswas , Sutirtha Paul , Joseph Agada , Matthias Thamm , Adrian Del Maestro This is my paper

Pith reviewed 2026-06-28 19:15 UTC · model grok-4.3

classification ⚛️ physics.comp-ph cond-mat.str-elcs.LGquant-ph
keywords Gaussian processphysical constraintsensemble modelingheteroskedastic noisequantum many-body systemsBose-Hubbard modelDMRGQMC
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The pith

Physically constrained ensemble Gaussian processes model noisy quantum simulations with better accuracy and physical consistency than standard approaches.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a framework to model expensive quantum many-body simulations that produce data with varying noise levels by extending Gaussian processes with physical constraints. It adds a weighted penalty term to the loss function to discourage unphysical predictions and trains an ensemble of such models whose outputs are combined using numerical quadrature to handle the heteroskedastic noise. Demonstrations on DMRG data for the Bose-Hubbard model predict the critical interaction strength for the superfluid-Mott transition, while QMC data for a confined quantum liquid guides optimization of conditions for one-dimensional superfluidity. The central goal is to enable reliable exploration of large parameter spaces without running prohibitive numbers of costly simulations. A sympathetic reader would care because the resulting surrogates could accelerate discovery of quantum phases and material properties by producing predictions that both fit the data and obey known physical rules.

Core claim

The pc-EGP framework enforces physical constraints as a user controlled weighted penalty to the data-driven loss function of the Gaussian Process (GP) surrogates. Then an ensemble of such GP models is trained with variable noisy simulations via numerical quadrature method where these multiple GP(s) at different nodes is integrated as a quadrature weighted average. Applied first to synthetic data and then to DMRG simulations of the Bose-Hubbard Model for predicting the critical Uc and to QMC simulations of a quantum liquid in a nanoporous silicate for optimizing a chemical environment, the method achieves a better balance of accuracy and physically meaningful predictions than conventional GP.

What carries the argument

Physically Constrained Ensemble Gaussian Process (pc-EGP) that adds a user-controlled weighted penalty for physical consistency to the GP loss function and integrates an ensemble of GPs trained on noisy data via numerical quadrature weighting.

If this is right

  • The critical interaction parameter Uc for the superfluid-to-Mott-insulator transition can be estimated reliably from limited DMRG runs.
  • Chemical environments realizing one-dimensional superfluidity can be optimized from fewer QMC simulations while respecting physical laws.
  • Expensive quantum many-body calculations with heteroskedastic noise can be replaced by surrogates that remain physically plausible over unsampled regions.
  • Parameter-space exploration of quantum systems becomes feasible with sparsely sampled, noisy simulation data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same penalty-plus-quadrature construction could be applied to other expensive simulation families if suitable physical constraints are supplied.
  • An automated procedure for selecting the penalty weight would remove a manual step that currently limits ease of use.
  • Direct comparisons against other physics-informed machine-learning surrogates would isolate the contribution of the ensemble quadrature step.

Load-bearing premise

Adding a user-controlled weighted penalty to the Gaussian process loss function enforces physical consistency without introducing new fitting artifacts or degrading predictive accuracy on the underlying data.

What would settle it

A case in which raising the penalty weight produces either predictions that violate the imposed physical constraints or measurably higher error on held-out simulation points would falsify the premise.

Figures

Figures reproduced from arXiv: 2606.11240 by Adrian Del Maestro, Arpan Biswas, Joseph Agada, Matthias Thamm, Sutirtha Paul.

Figure 1
Figure 1. Figure 1: Performance of pc-GP over synthetic test function [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Performance of pc-GP, EGP and pc-EGP for a synthetic test function with heteroskedastic noise. The test mean function is the same as fig 1(a) where the details on the expansion to the noisy function is provided in Supplementary Material Fig. A1. Figs 2(a) - 1(d) are the prediction (blue solid lines) from GP, pc-GP, EGP and pc-EGP respectively with 40 randomly selected training samples. For pc-EGP and pc-GP… view at source ↗
Figure 3
Figure 3. Figure 3: shows the performance of the estimation of the proposed model pc-EGP, in learning the Bose–Hubbard model, leveraging DMRG and QMC simulations, to predict the critical interaction parameter 𝑈𝑐/𝐽 governing the superfluid-to-Mott-insulator transition where (1 − 𝜁( 𝑈 𝐽 )) 2 = 0. For this dataset, the noise heteroskedastic as each simulated value of 𝑈/𝐽 has a different propagated uncertainty. Moreover, while th… view at source ↗
Figure 5
Figure 5. Figure 5: a shows the expensive QMC simulated ground truth of density 𝜌 at nanopore radius 𝑅 = 6.0Å over the 2D parameter space of radial position 𝑟 and chemical potential 𝜈. Fig. 5b shows the respective QMC simulation noise (variance), where we can see the higher density region has higher noise. From domain-knowledge, we know that at any value of (𝑟, 𝜈), 𝜌 ≥ 0. Out of the 5800 simulated data (each value of 𝜐 corres… view at source ↗
Figure 6
Figure 6. Figure 6: Performance of pc-EGP over noisy QMC simulations of helium confined inside cylindrical nanopores with different filling fractions set by the tunable chemical potential 𝜐. Simulation details are included in Ref. 35 . Here, we consider a 2D parameter space of radial position 𝑟 and chemical potential 𝜈 and the function space of density 𝜌 where the physical constraint on the density is 𝜌 ≥ 0 [PITH_FULL_IMAGE:… view at source ↗
read the original abstract

Accurate modeling of quantum many-body systems often requires computationally expensive simulations such as Density Matrix Renormalization Group (DMRG) or Quantum Monte Carlo (QMC) calculations. These methods, while precise, impose significant time and resource constraints, limiting their use in exhaustive parameter exploration. Moreover, these expensive simulations can contain variable errors over the large unknown parameter space, which needs to be quantified and propagated. Thus, predictive modelling is required to estimate the functional space accurately over scarcely sampled data with heteroskedastic noise, while preserving the physical relevance of the estimation. Therefore, we present a Physically Constrained Ensemble Gaussian Process (pc-EGP) framework designed to efficiently model complex and noisy quantum systems under physical consistency constraints. The proposed method first enforces physical constraints as a user controlled weighted penalty to the data-driven loss function of the Gaussian Process (GP) surrogates. Then an ensemble of such GP models is trained with variable noisy simulations via numerical quadrature method where these multiple GP(s) at different nodes is integrated as a quadrature weighted average. We first demonstrate the framework on synthetically generated data before applying to quantum systems. In the first case study, we leverage DMRG simulations of the Bose-Hubbard Model to predict the critical interaction parameter Uc governing the superfluid-to-Mott-insulator transition. In the second case study, we demonstrate our method on QMC simulations, of a quantum liquid confined inside a nanoporous silicate with the goal of optimizing a chemical environment to realize a one-dimensional superfluid. Compared to conventional GP, pc-EGP achieves a better balance of accuracy and physically meaningful predictions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a Physically Constrained Ensemble Gaussian Process (pc-EGP) framework for surrogate modeling of expensive quantum many-body simulations (DMRG and QMC) that exhibit heteroskedastic noise. Physical constraints are incorporated by adding a user-controlled weighted penalty term to the standard GP loss; an ensemble of such GPs is then trained on noisy data and combined via numerical quadrature to produce integrated predictions. The method is first validated on synthetic data and then applied to (i) DMRG data for the Bose-Hubbard model to predict the critical interaction Uc of the superfluid-Mott transition and (ii) QMC data for a quantum liquid in a nanopore to optimize conditions for a one-dimensional superfluid. The central claim is that pc-EGP achieves a superior accuracy-versus-physical-consistency trade-off relative to conventional GP.

Significance. If the central claim is substantiated, the pc-EGP construction would provide a practical surrogate-modeling tool for quantum many-body problems where direct simulation is prohibitive and physical consistency must be preserved. The combination of a penalty-based constraint mechanism with quadrature-based ensemble integration for heteroskedastic noise is a technically coherent extension of existing GP methods and could accelerate parameter-space exploration in condensed-matter applications.

major comments (2)
  1. [Method description] Method description (abstract and §2): the framework adds physical constraints 'as a user controlled weighted penalty' to the GP loss, yet supplies no procedure for selecting or validating the penalty weight, no ablation study over weight values, and no trade-off curves relating constraint violation to predictive error on the DMRG or QMC data. Without this quantitative evidence the claim that pc-EGP 'achieves a better balance of accuracy and physically meaningful predictions' cannot be assessed and may reflect manual tuning rather than an intrinsic property of the construction.
  2. [Results (case studies)] Results sections (case studies 1 and 2): the reported improvements over conventional GP are stated without accompanying error bars, statistical significance tests, or explicit comparison tables that isolate the contribution of the penalty term versus the ensemble quadrature component. This omission leaves the load-bearing claim of improved accuracy-plus-consistency unsupported by the presented evidence.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from a concise statement of the precise physical constraints enforced in each case study (e.g., positivity of density, monotonicity of energy, etc.).
  2. [Method] Notation for the quadrature weights and the ensemble integration formula should be introduced with an equation number to facilitate reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and agree that the manuscript would benefit from additional quantitative support for the penalty-weight selection and for the performance claims. Revisions will be made accordingly.

read point-by-point responses

  1. Referee: [Method description] Method description (abstract and §2): the framework adds physical constraints 'as a user controlled weighted penalty' to the GP loss, yet supplies no procedure for selecting or validating the penalty weight, no ablation study over weight values, and no trade-off curves relating constraint violation to predictive error on the DMRG or QMC data. Without this quantitative evidence the claim that pc-EGP 'achieves a better balance of accuracy and physically meaningful predictions' cannot be assessed and may reflect manual tuning rather than an intrinsic property of the construction.

    Authors: We agree that the current manuscript lacks an explicit procedure for penalty-weight selection and the associated ablation/trade-off analyses. In the revised version we will add (i) a cross-validation-based protocol for choosing the weight that balances data fidelity against constraint violation, (ii) ablation results over a range of weights, and (iii) trade-off curves of predictive error versus constraint violation for both the DMRG and QMC case studies. These additions will allow readers to assess whether the reported balance is systematic rather than the result of manual tuning. revision: yes

  2. Referee: [Results (case studies)] Results sections (case studies 1 and 2): the reported improvements over conventional GP are stated without accompanying error bars, statistical significance tests, or explicit comparison tables that isolate the contribution of the penalty term versus the ensemble quadrature component. This omission leaves the load-bearing claim of improved accuracy-plus-consistency unsupported by the presented evidence.

    Authors: We acknowledge that the present results do not include error bars, statistical tests, or component-isolating tables. The revised manuscript will report error bars on all quantitative metrics, include paired statistical significance tests (e.g., Wilcoxon or t-tests) between pc-EGP and baseline GP, and add comparison tables that separately quantify the effect of the penalty term and the quadrature ensemble integration on both accuracy and physical-consistency metrics. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The pc-EGP method is presented as a standard modeling procedure: a user-weighted penalty term is added to the GP loss to enforce physical constraints, followed by ensemble training via numerical quadrature on noisy simulation data. No equation or step in the abstract or method description reduces a reported prediction or result to a quantity already fitted to the same data by construction. No self-citations are invoked as load-bearing uniqueness theorems, no ansatz is smuggled via prior work, and no fitted parameter is relabeled as an independent prediction. The comparison to conventional GP is framed as an empirical demonstration on DMRG and QMC data rather than a definitional identity. This places the work in the normal non-circular category.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted. The method implicitly assumes that physical constraints can be expressed as differentiable penalty terms and that quadrature nodes adequately sample the heteroskedastic noise distribution, but these are not quantified.

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Reference graph

Works this paper leans on

51 extracted references · 4 canonical work pages · 1 internal anchor

  1. [1]

    To accelerate the search for optimal parameter regimes, much effort has been focused on designing autonomous workflows via machine learning (ML)-driven selection (active learning)

    INTRODUCTION The modern era of scientific discovery brings forth the challenge of exploring complex, large and time-consuming multidimensional domain specific parameter and function spaces. To accelerate the search for optimal parameter regimes, much effort has been focused on designing autonomous workflows via machine learning (ML)-driven selection (acti...

  2. [2]

    These quantum phases can be defined as a function of several tuning parameters of the microscopic model

    are required to explore transitions between different quantum phases. These quantum phases can be defined as a function of several tuning parameters of the microscopic model. In order to explore over these tuning parameters to find an optimal point where a phase transition occurs , it may require millions of CPU core hours even to tune a single parameter ...

  3. [3]

    METHODOLOGY The proposed pc -EGP model has two key developments -

  4. [4]

    the injection of physical constraints during the training process by jointly optimizing the model hyper parameters via a user-controlled integrated physical loss function, and

  5. [5]

    an ensemble modelling approach via numerical quadrature method to propagate and quantify heteroskedastic noise present in physical systems. 2.1. Physical Loss Integration into a Standard Gaussian Process (GP) model: The general form of the GP model is as follows: 𝑦(𝑥) = ∆(𝑥)+ 𝜀 (1) where 𝜀~𝑁(0, 𝜎𝑒 2𝐼) is the standard fixed noise model with zero mean and v...

  6. [6]

    RESULTS From the above analysis over the synthetically generated functions, we can see promising results where the proposed method has the ability 1) to rectify the estimation from the enormous sampled data, 2) gaining higher confidence in estimation from the data with heteroskedastic noise and 3) balancing the physical and data -driven constraints for an...

  7. [7]

    We consider the Matern Kernel eq. 5. From fig. 6a, we can see while the standard GP prediction matches best with the ground truth QMC simulation in Fig. 5a , it provides unphysical function predictions over as much as 38% of the overall parameter space. Thus, to define a better performance metric , we penalize the mean absolute error with constraint viola...

  8. [8]

    Thus, to balance both, it appears preferable to choose 𝑤1 = 2

    As the overall prediction error increases, we see the penalized Mean Square Errors, 𝑝𝑀𝐴𝐸, for these 3 cases are 0.035, 0.033 and 0.032 respectively. Thus, to balance both, it appears preferable to choose 𝑤1 = 2. However, as preferred by the domain expert objectives, the se weights can be controlled which aids the human -in-the loop for alignment in ML-dri...

  9. [9]

    CONCLUSION To summarize, we have proposed a physically-constrained ensemble Gaussian Process model for improved and physically meaningful learning of computationally expensive quantum systems. The two key developments in the proposed approach, during the model training process, are, designing 1) a pathway for inclusion of physical constraint validation an...

  10. [10]

    & Del Maestro, A

    Barghathi, H., Usadi, C., Beck, M. & Del Maestro, A. Compact unary coding for bosonic states as efficient as conventional binary encoding for fermionic states. Phys. Rev. B 105, L121116 (2022)

    2022

  11. [11]

    The density-matrix renormalization group in the age of matrix product states

    Schollwöck, U. The density-matrix renormalization group in the age of matrix product states. Ann. Phys. 326, 96–192 (2011)

    2011

  12. [12]

    White, S. R. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863– 2866 (1992)

    1992

  13. [13]

    Thamm, M. et al. Berezinskii-Kosterlitz-Thouless Renormalization Group Flow at a Quantum Phase Transition. Phys. Rev. Lett. 135, 116002 (2025)

    2025

  14. [14]

    Casiano-Diaz, E., Herdman, C. M. & Del Maestro, A. A path integral ground state Monte Carlo algorithm for entanglement of lattice bosons. SciPost Phys. 14, 054 (2023)

    2023

  15. [15]

    & Sorella, S

    Becca, F. & Sorella, S. Quantum Monte Carlo Approaches for Correlated Systems. (Cambridge University Press, 2017)

    2017

  16. [16]

    Shahriari, B., Swersky, K., Wang, Z., Adams, R. P. & de Freitas, N. Taking the Human Out of the Loop: A Review of Bayesian Optimization. Proc. IEEE 104, 148–175 (2016)

    2016

  17. [17]

    R., Schonlau, M

    Jones, D. R., Schonlau, M. & Welch, W. J. Efficient Global Optimization of Expensive Black-Box Functions. J. Glob. Optim. 13, 455–492 (1998)

    1998

  18. [18]

    & Hoyle, C

    Biswas, A. & Hoyle, C. An Approach to Bayesian Optimization for Design Feasibility Check on Discontinuous Black-Box Functions. J. Mech. Des. 143, (2021)

    2021

  19. [19]

    Quadrianto, N., Kersting, K. & Xu, Z. Gaussian Process. in Encyclopedia of Machine Learning (eds. Sammut, C. & Webb, G. I.) 428–439 (Springer US, Boston, MA, 2010). doi:10.1007/978-0-387-30164- 8_324

    work page doi:10.1007/978-0-387-30164- 2010

  20. [20]

    Deringer, V. L. et al. Gaussian Process Regression for Materials and Molecules. Chem. Rev. 121, 10073–10141 (2021)

    2021

  21. [21]

    Noack, M. M. et al. Autonomous materials discovery driven by Gaussian process regression with inhomogeneous measurement noise and anisotropic kernels. Sci. Rep. 10, 17663 (2020)

    2020

  22. [22]

    Brochu, E., Cora, V. M. & de Freitas, N. A Tutorial on Bayesian Optimization of Expensive Cost Functions, with Application to Active User Modeling and Hierarchical Reinforcement Learning. Preprint at https://doi.org/10.48550/arXiv.1012.2599 (2010)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1012.2599 2010

  23. [23]

    Cox, D. D. & John, S. A statistical method for global optimization. in [Proceedings] 1992 IEEE International Conference on Systems, Man, and Cybernetics 1241–1246 vol.2 (1992). doi:10.1109/ICSMC.1992.271617

    work page doi:10.1109/icsmc.1992.271617 1992

  24. [24]

    Jones, D. R. A Taxonomy of Global Optimization Methods Based on Response Surfaces. J. Glob. Optim. 21, 345–383 (2001)

    2001

  25. [25]

    Kushner, H. J. A New Method of Locating the Maximum Point of an Arbitrary Multipeak Curve in the Presence of Noise. J. Basic Eng. 86, 97–106 (1964)

    1964

  26. [26]

    Sanchez, S. L. et al. Physics-driven discovery and bandgap engineering of hybrid perovskites. Digit. Discov. 3, 1577–1590 (2024)

    2024

  27. [27]

    Chang, J. et al. Efficient Closed-loop Maximization of Carbon Nanotube Growth Rate using Bayesian Optimization. Sci. Rep. 10, 9040 (2020)

    2020

  28. [28]

    D., Hou, Z., Mizoguchi, T

    Ueno, T., Rhone, T. D., Hou, Z., Mizoguchi, T. & Tsuda, K. COMBO: An efficient Bayesian optimization library for materials science. Mater. Discov. 4, 18–21 (2016)

    2016

  29. [29]

    V., Ziatdinov, M

    Kalinin, S. V., Ziatdinov, M. & Vasudevan, R. K. Guided search for desired functional responses via Bayesian optimization of generative model: Hysteresis loop shape engineering in ferroelectrics. J. Appl. Phys. 128, 024102 (2020). 12

    2020

  30. [30]

    N., Ziatdinov, M., Eliseev, E

    Biswas, A., Morozovska, A. N., Ziatdinov, M., Eliseev, E. A. & Kalinin, S. V. Multi-objective Bayesian optimization of ferroelectric materials with interfacial control for memory and energy storage applications. J. Appl. Phys. 130, 204102 (2021)

    2021

  31. [31]

    Morozovska, A. N. et al. Chemical control of polarization in thin strained films of a multiaxial ferroelectric: Phase diagrams and polarization rotation. Phys. Rev. B 105, 094112 (2022)

    2022

  32. [32]

    N., Eliseev, E

    Morozovska, A. N., Eliseev, E. A., Biswas, A., Morozovsky, N. V. & Kalinin, S. V. Effect of Surface Ionic Screening on Polarization Reversal and Phase Diagrams in Thin Antiferroelectric Films for Information and Energy Storage. Phys. Rev. Appl. 16, 044053 (2021)

    2021

  33. [33]

    Tao, S., van Beek, A., Apley, D. W. & Chen, W. Multi-Model Bayesian Optimization for Simulation- Based Design. J. Mech. Des. 143, (2021)

    2021

  34. [34]

    & Ziatdinov, M

    Narasimha, G., Hus, S., Biswas, A., Vasudevan, R. & Ziatdinov, M. Autonomous convergence of STM control parameters using Bayesian optimization. APL Mach. Learn. 2, 016121 (2024)

    2024

  35. [35]

    Liu, Y. et al. Experimental discovery of structure– property relationships in ferroelectric materials via active learning. Nat. Mach. Intell. 4, 341–350 (2022)

    2022

  36. [36]

    & Hernández-Lobato, J

    Griffiths, R.-R. & Hernández-Lobato, J. M. Constrained Bayesian optimization for automatic chemical design using variational autoencoders. Chem. Sci. 11, 577–586 (2020)

    2020

  37. [37]

    Burger, B. et al. A mobile robotic chemist. Nature 583, 237–241 (2020)

    2020

  38. [38]

    Kusne, A. G. et al. On-the-fly closed-loop materials discovery via Bayesian active learning. Nat. Commun. 11, 5966 (2020)

    2020

  39. [39]

    Dave, A. et al. Autonomous optimization of non- aqueous Li-ion battery electrolytes via robotic experimentation and machine learning coupling. Nat. Commun. 13, 5454 (2022)

    2022

  40. [40]

    Harris, S. B. et al. Autonomous Synthesis of Thin Film Materials with Pulsed Laser Deposition Enabled by In Situ Spectroscopy and Automation. Small Methods n/a, 2301763

  41. [41]

    Bayesian Optimization

    Garnett, R. Bayesian Optimization. Cambridge Core https://www.cambridge.org/core/books/bayesian- optimization/11AED383B208E7F22A4CE1B5BCB ADB44 (2023) doi:10.1017/9781108348973

    work page doi:10.1017/9781108348973 2023

  42. [42]

    Balandat, M. et al. BOTORCH: a framework for efficient monte-carlo Bayesian optimization. in Proceedings of the 34th International Conference on Neural Information Processing Systems 21524– 21538 (Curran Associates Inc., Red Hook, NY, USA, 2020)

    2020

  43. [43]

    Ziatdinov, M. A. et al. Hypothesis Learning in Automated Experiment: Application to Combinatorial Materials Libraries. Adv. Mater. 34, 2201345 (2022)

    2022

  44. [44]

    Paul, S., Lakoba, T., Sokol, P. E. & Del Maestro, A. Localization and wetting of $^{4}\mathrm{He}$ inside preplated nanopores. Phys. Rev. B 113, 075433 (2026)

    2026

  45. [45]

    S., Prisk, T

    Nichols, N. S., Prisk, T. R., Warren, G., Sokol, P. & Del Maestro, A. Dimensional reduction of helium-4 inside argon-plated MCM-41 nanopores. Phys. Rev. B 102, 144505 (2020)

    2020

  46. [46]

    & Affleck, I

    Del Maestro, A., Boninsegni, M. & Affleck, I. $^{4}\mathrm{He}$ Luttinger Liquid in Nanopores. Phys. Rev. Lett. 106, 105303 (2011). 13 Supplementary Materials of the paper titled Physically Constrained Ensemble Gaussian Process Modelling for Expensive Quantum Systems with Heteroskedastic Noise Arpan Biswas1,2, Sutirtha Paul2,3, Joseph Agada4, Matthias Tha...

    2011

  47. [47]

    Randomly select 𝑚 samples from the parameter space 𝑿

    Initialization for BO: State maximum BO iteration, 𝑀. Randomly select 𝑚 samples from the parameter space 𝑿. Assuming 𝑓 is the expensive objective function. Set 𝑘 = 1. Evaluate 𝑚 samples for objective as, 𝒀(𝑿). Build training data matrices, 𝑫𝒌 = {𝑿𝒌, 𝒀𝒌}. For 𝑘 ≤ 𝑀

  48. [48]

    Here, we have integrated our proposed pc-EGP model

    Surrogate Modelling: Develop or update GP models, given the training data, as ∆(𝑫𝒌). Here, we have integrated our proposed pc-EGP model

  49. [49]

    Here, 𝒋 GP models of the pc - EGP provide parallel estimations and then we calculate the weighted mean and variances as per eq

    Posterior Predictions : Given the surrogate model, compute posterior means and variances for the unexplored locations, 𝑿𝒌̂, as 𝝁(𝒀(𝑿𝒌̂ )|∆𝒋 and 𝝈𝟐(𝒀(𝑿𝒌̂ )|∆𝒋 respectively. Here, 𝒋 GP models of the pc - EGP provide parallel estimations and then we calculate the weighted mean and variances as per eq. 19-20

  50. [50]

    Acquisition function: Compute and maximize acquisition function, max 𝑥𝑏𝑒𝑠𝑡∈𝑿𝒌̂ Υ(𝑓|∆) to select next best location, 𝒙𝒃𝒆𝒔𝒕 for evaluations

  51. [51]

    Augment data, 𝐷𝑘+1 = [𝐷𝑘; {𝒙𝒃𝒆𝒔𝒕, 𝑦}

    Augmentation: Evaluate 𝑦(𝒙𝒃𝒆𝒔𝒕). Augment data, 𝐷𝑘+1 = [𝐷𝑘; {𝒙𝒃𝒆𝒔𝒕, 𝑦} . Repeat Step 2 -5 till convergence. 14 Appendix B. Additional Figures Figure A1. Test function with heteroskedastic noise as denoted from the shaded region. The red dots are the sampled training data. Figure A2. Additional Analysis: Performance of pc-EGP over noisy QMC simulations of t...