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Bayesian Optimisation under State-Preservation Constraints

Gabriel Diaz-Aylwin, Vignesh Gopakumar, Omkar Myatra, David Moulton, Lorenzo Zanisi, David S. Leslie, Henry B. Moss

arXiv:2610.12150v1cs.LG

TL;DR

Engineering design must improve established designs while preserving selected state observables, but constrained BO can struggle when feasible regions are thin and anisotropic. The paper precomputes and refines a linearised feasible-set geometry for BO proposals. In divertor optimisation, this approach reaches a median detachment threshold 5–7× below the best competitor in the currents-only problem while keeping about 90% of evaluations within a 0.5 cm boundary tolerance.

  • Problem

    State-preservation constraints can produce thin, anisotropic feasible sets that are difficult to learn from a handful of expensive evaluations.

    Source: §1 · p. 1

  • Method

    The method precomputes an ellipsoidal approximation of the feasible set from the linearised constraint response and refines its response map online for BO proposals.

    Source: §1 · pp. 1–2

  • Results

    The median detachment threshold is 5–7× below the best competitor in the currents-only problem, with about 90% of evaluations within a 0.5 cm boundary tolerance.

    Source: §6 · p. 9

  • Takeaways & Limitations

    The proposal geometry is a self-contained BO module that can be plugged into other BO advances, including batch, multi-fidelity, multi-task, and high-dimensional methods.

    Source: §6 · p. 9

    Direct simulator-constrained optimisation is sensible only when feasibility checks are significantly cheaper than objective evaluation; gradients are not straightforward in this setting.

    Source: §2 · p. 3

Abstract

from arXiv · show

In many engineering design problems, the objective and constraints depend on the state: the solution of a PDE determined by the design parameters. We consider improving a design while holding selected state observables near trusted values, which we call state preservation constraints. Constrained Bayesian optimisation handles these with a learnt feasibility model, but struggles with this problem's highly anisotropic feasible set. Our central idea is to pre-compute the set of controls whose linearised constraint response stays within tolerance, thereby pulling back the state-space constraint into design space. This linearisation defines an ellipsoid from which we can efficiently draw a large number of well-spread candidates. The underlying linear response map is refined online, and the ellipsoid is rebuilt accordingly. We demonstrate the method end-to-end on our key application - Tokamak divertor optimisation under plasma-boundary preservation.

1 INTRODUCTION

Engineering design often relies on expensive evaluations and must improve an established design without losing its desirable properties. The paper proposes an ellipsoidal, constraint-aware proposal geometry for Bayesian optimisation and demonstrates it on tokamak divertor design.

1 INTRODUCTION

  • Bayesian optimisation suits engineering settings with expensive objective evaluations, including legacy nondifferentiable codebases.The introduction situates the approach within a longer history of engineering optimisation.Source: §1 · p. 1
  • State-preservation constraints can create a thin, anisotropic feasible set, leaving most proposed designs infeasible and difficult to learn from a handful of expensive scalar observations.The problem arises when improvements must preserve desirable properties of an established baseline.Source: §1 · p. 1
  • The method precomputes a first-order approximation of the feasible set and uses the resulting ellipsoid as a constraint-aware proposal geometry in Bayesian optimisation.Millions of well-spread candidates can be drawn from the closed-form region at negligible cost, and its linear response map is refined online.Source: §1 · pp. 1–2
  • Figure 1 contrasts uniform design-space samples with samples from the linearised feasible region and shows the associated PDE solutions relative to the baseline.The displayed full domain has d = 10 current variables.Source: Figure 1 · p. 2
  • The method demonstrates substantial sample-efficiency gains over standard constrained-BO baselines for divertor optimisation, with ablations identifying proposal geometry as the principal source of improvement.The design objective is improved heat-exhaust performance while preserving the baseline plasma boundary, or LCFS.Source: §1 · p. 2

2 BACKGROUND

Bayesian optimisation uses probabilistic surrogates and acquisition functions to allocate evaluations when objectives are expensive. The section reviews constraint surrogates and local feasible-set approximations, and explains why direct simulator-constrained optimisation is not used here.

Bayesian optimisation

  • Bayesian optimisation builds a probabilistic surrogate from observed evaluations and selects the next design by maximising an acquisition function balancing predicted objective value and uncertainty.Gaussian processes provide a closed-form posterior predictive distribution in the described setup.Source: §2 · p. 2

Constrained Bayesian optimisation

  • Standard constrained BO learns probabilistic constraint surrogates to estimate feasibility, but thin, anisotropic state-preservation regions can be hard to resolve from few expensive evaluations.Constrained expected improvement is one example; alternatives include information-theoretic acquisitions and penalty formulations.Source: §2 · pp. 2–3

Multi-objective Bayesian optimisation

  • Multi-objective BO seeks improvements over the Pareto front of optimal tradeoffs, commonly using Expected Hyper-Volume Improvement as its acquisition function.The passage names implementations by Daulton, Balandat, and Bakshy (2020).Source: §2 · p. 3

Local feasible-set approximations

  • The paper uses local linearised feasible-set geometry for BO proposals and updates that approximation online, adapting ideas from constrained optimisation.Related approaches include sequential quadratic programming, trust-region variants, and derivative-free local constraint models.Source: §2 · p. 3

Optimising under the simulator constraint

  • Direct simulator-defined feasibility is considered unsuitable unless feasibility checks are significantly cheaper than objective evaluation; gradients are also not straightforward here.The resulting non-convex acquisition optimisation can be local and sensitive to initialisation.Source: §2 · p. 3

3 CONSTRAINT-AWARE CANDIDATE GENERATION

The method approximates preservation around a baseline with a first-order feasible region for Bayesian-optimisation proposals. It samples that region broadly, refines its response map online, and models residual observable changes to account for linearisation error.

3 CONSTRAINT-AWARE CANDIDATE GENERATION

  • The task optimises F(ψ(x)) subject to observable-change violation v(x) ≤ τ, relative to baseline design x0.Source: §3 · p. 3

3.1 The linearised feasible region

  • The first-order preservation constraint pulls the observable-space norm back through J as M = J⊤GJ, defining an ellipsoidal feasible region E.The proposal geometry uses E∩X, which remains bounded when M is singular.Source: §3.1 · p. 3

3.2 Ray-based intersection sampling and maximin filtering

  • Ray-based sampling generates a large candidate pool inside E∩B, then maximin filtering retains a well-spread subset.B is the engineering domain or the current objective-focused trust region.Source: §3.2 · p. 4

3.3 Online refinement

  • Each simulator evaluation provides response data at no extra solver cost; least squares refits the linear map so the proposal geometry adapts to finite-scale behaviour.The finite-difference Jacobian supplies the initial geometry before BO data arrive.Source: §3.3 · p. 4

3.4 Feasibility beyond the linearisation

  • To address infeasible edge points from linearisation error, the method models observable-change fields with a low-rank residual basis and GPs on its coefficients.It estimates norm-based feasibility through a moment-matched generalised χ2 tail; the field model outperforms a scalar GP on the norm in Section 5.Source: §3.4 · p. 4

4 DIVERTOR OPTIMISATION

The application optimises tokamak divertor design while preserving the baseline plasma boundary. The paper describes the magnetic design and LCFS constraint, then positions its norm-based preservation construction among fusion-control approaches.

4 DIVERTOR OPTIMISATION

  • The design uses poloidal-field coil currents, later including coil positions, while the state is the poloidal flux function returned by the Grad–Shafranov solver.The observable is the LCFS shape, whose changes are represented by signed normal displacements and whose preservation is measured by RMS displacement.Source: §4 · pp. 4–5
  • The divertor objective is to lower the detachment-threshold impurity fraction, making detachment easier to access and widening the operating window.The threshold is defined at fixed upstream conditions.Source: §4 · p. 5

4.1 Related work in Fusion Research

  • Virtual circuits use linearised coil-to-boundary response for shape control; this work uses the principle to construct a search region instead.Relaxing the response null space to a region would require a shape-parameter norm; the method instead uses the L2 norm of LCFS normal displacement.Source: §4.1 · p. 5
  • The norm-based constraint decouples LCFS resolution from design dimension, resolving the boundary at O(10^4) points while retaining all d design directions.The tolerance τ controls the allowed displacement.Source: §4.1 · p. 6
  • Related approaches either constrain leading spherical-harmonic coefficients, sacrificing a degree of freedom per added harmonic, or pin flux at sampled separatrix points without constraining between them.The harmonic approach fixes the field throughout a source-free ball containing the plasma rather than directly preserving the LCFS.Source: §4.1 · p. 6

5 EXPERIMENTS

The experiments compare linearised-feasible-set proposals with a BO competitor on currents-only and joint current-position divertor optimisation, then extend the approach to multiple objectives. The method performs better in the reported comparisons, while online refinement helps maintain feasibility as the response becomes more nonlinear.

5 EXPERIMENTS

  • Experiments use a shared MAST-U-like baseline with 0.5 cm LCFS-displacement tolerance; budgets are 1,000 evaluations for currents-only and 1,500 for joint optimisation.The runs report 50 seeds; initial response-map construction requires 2d additional finite-difference solves.Source: §5 · p. 6
  • The algorithm draws candidates from the linearised feasible region intersected with a trust region, refits the response map, rebuilds the region, and updates its surrogates.Candidate pools are maximin-filtered, and feasibility is assessed with a field model.Source: §5 · p. 6

5.1 Experiment 1: currents only (d=10)

  • The proposal geometry is the main source of currents-only gains; the field model raises feasibility to 89.5%, while excursions beyond the linearised region add little objective improvement.Random sampling from E∩B without a surrogate reaches median final cZ 0.018 at 90.8% feasibility; hard confinement alone gives 77% feasibility, versus 47% for unconstrained scalar-GP refinement.Source: §5.1 · p. 7
  • Freezing the response map costs roughly 12% feasibility without significant objective effect; trust-region removal does not hurt the method but makes the competitor a non-starter.For the competitor, larger initial trust regions buy progress at the cost of feasibility; at L ≥ 0.015, feasibility falls below 8% and progress stalls near cZ ≈ 0.065.Source: §5.1 · p. 7
  • At 1,000 evaluations, the method reaches median final cZ 0.012, 5.2× better than the best competitor, with 89.5% of evaluations feasible.The comparison covers the competitor sweep and reports final performance for the currents-only task.Source: §5.1 · pp. 7–8, Figure 3 · p. 7, Table 1 · p. 7

5.2 Experiment 2: include positions (d=30)

  • In the combined currents-and-position optimization, our method is more sample-efficient than competitor C1–C8, achieving a 2.1× lower detachment threshold at the same evaluation budget.The combined optimization has a much more nonlinear control-to-LCFS perturbation response than the currents-only case.Source: §5.2 · p. 8, Figure 4 · p. 8
  • With positions included, online refitting cuts held-out median violation-prediction error from 1.07 cm to 0.031 cm and retains 83.3% feasibility.The frozen linearisation's median error exceeds the 0.5 cm tolerance.Source: §5.2 · p. 8

5.3 Extension 3: an additional objective

  • The same proposal geometry supports multi-objective optimisation: the demonstration finds dense Pareto fronts for currents-only and joint current-position designs using EHVI.The two objectives are connection length and total flux expansion; no competitor is run because a multi-objective trust region would be nontrivial.Source: Figure 5 · p. 8, §5.3 · p. 8

6 CONCLUSION

The proposed geometry addresses thin, anisotropic preservation constraints by precomputing and refining a first-order feasible-set approximation. On the divertor problem, it reaches lower median detachment thresholds than the best competitor while keeping most evaluations within the boundary tolerance.

6 CONCLUSION

  • The currents-only method reaches a median detachment threshold 5–7× below the best competitor while holding ∼90% of evaluations within a 0.5 cm boundary tolerance.Uniform sampling of the d = 10 current box produced no feasible designs in 1,000 solves.Source: §6 · p. 9
  • The method precomputes the first-order feasible-set approximation from 2d solves, then refines it online to address a thin, anisotropic constraint set.Source: §6 · p. 9
  • The authors present the approach as a general framework for tightly constrained engineering BO and report that its module can be plugged into orthogonal BO advances, including MOBO.Future applications named include radio-frequency cavity and stellarator design.Source: §6 · p. 9

A HYPERPARAMETERS

Table 2 records the numerical settings used in the experiments, spanning currents-only and joint currents-and-positions problems.

A HYPERPARAMETERS

  • Table 2 lists every numerical setting used in the experiments.Source: A HYPERPARAMETERS · p. 13, Table 2 · p. 13
  • The settings cover the currents-only problem with d = 10 and the joint currents-and-positions problem with d = 30.Source: Table 2 · p. 13
  • The table’s scope is experimental settings for two problem formulations: currents-only and jointly optimized currents and positions.Source: Table 2 · p. 13

B MEASURING THE LCFS DISPLACEMENT

The LCFS displacement is measured along outward normals at uniformly arclength-resampled points on the baseline contour. Finite differences of these displacements form the linear response matrix used for the preservation constraints.

B MEASURING THE LCFS DISPLACEMENT

C SAMPLING DETAILS

Candidates are drawn along rays within the intersection of the ellipsoid and box. The sampling combines feasible origins, constraint-aware and Euclidean directions, and bounded ray lengths before maximin down-selection.

C SAMPLING DETAILS

  • Candidate rays start from the baseline x0 and incumbent xbest when they lie in E ∩ B, splitting candidates evenly between available origins.Fallback origins are tried in order: the incumbent under its proposing region snapshot, a point near the ellipsoid boundary toward xbest, then x0 with the box trust region dropped for that iteration.Source: C SAMPLING DETAILS · p. 14
  • Half the directions are sampled uniformly in the constraint-induced ellipsoid, concentrating along coil combinations that barely move the LCFS; the other half are uniform Euclidean directions.Source: C SAMPLING DETAILS · p. 14
  • Each ray length is sampled uniformly from zero to the nearer ellipsoid or box boundary, and the candidate pool is reduced by greedy maximin.Source: C SAMPLING DETAILS · p. 14
  • The ray-sampling scheme is heuristic, and its origin-versus-direction split depends on the preservation constraints.It is fast and parallelisable, enabling good coverage after maximin down-sampling.Source: C SAMPLING DETAILS · p. 14

D BILINEAR STRUCTURE OF THE EQUILIBRIUM RESPONSE

The coil contribution to flux is linear in coil currents, but perturbing both currents and positions introduces a bilinear interaction. The first-order response is linear in c = (δI, m), and the joint problem builds J and E in c before mapping them back to x.

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