Bayesian Optimisation under State-Preservation Constraints

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

cs.LGOct 8, 2026

Bayesian Optimisation under State-Preservation Constraints

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

Organizations: Lancaster University, UK · UK Atomic Energy Authority

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1: Left: (a) uniform samples across coil-current design variables $\mathcal{X}$ and (b) well-spread samples from the linearised feasible region $E$ , restricted to the D1, D2, D3 coils for presentation. Points feasible under the LCFS preservation constraint (desired fixed central region) are green, infeasible red. The starred point is our baseline solution. Right: (c, d) the resulting PDE solutions of the sampled design variables superposed on the baseline solution (black) corresponding to the designs in (a) and (b). This is a vertical slice of a tokamak reactor torus. (e, f) the same construction over the full $d=10$ current domain, showing how tightly $E$ constrains the LCFS. Adopting $E$ as the proposal geometry within BO is the essence of our method.

Panel of Figure 2 — Figure 2: (a) A reconstructed super-X equilibrium from MAST-U shot $53349$ . We take this equilibrium as our baseline (starting design). (b, c) Random current perturbations (within known MAST-U PF coil bounds) demonstrating the sensitivity of the LCFS (black traced contour). The centre of the black boxes denotes coil location. We will be optimising the current strengths (and later the coil locations) in order to get desirable properties of the separatrix (red contour line), whilst attempting to preserve the LCFS at its baseline shape.

Panel of Figure 3 — Figure 3: Currents-only ( $d=10$ ), 1000 evaluations. Median (bold), IQR (band) and every individual seed (faint). (a),(c) y-axis is best feasible $c_{Z}$ ; the dotted line is the baseline $c_{Z}=0.086$ . (b),(d) y-axis is cumulative feasible fraction. (a),(b) our method (A7) against the full competitor sweep C1–C8 over its trust-region size $L$ . (c),(d) the best competitor setting (C7) against every ablation A1–A8. Identifiers and colours are shared across all four panels and with Table 1 .

	Run	median $c_{Z}$	IQR	median feas.	IQR
C1	competitor, $L=0.001$	0.0827	[0.0787, 0.0837]	0.720	[0.646, 0.768]
C2	competitor, $L=0.005$	0.0702	[0.0665, 0.0728]	0.214	[0.155, 0.619]
C3	competitor, $L=0.01$	0.0683	[0.0651, 0.0723]	0.203	[0.083, 0.398]
C4	competitor, $L=0.015$	0.0692	[0.0653, 0.0717]	0.057	[0.045, 0.103]
C5	competitor, $L=0.02$	0.0646	[0.0563, 0.0722]	0.064	[0.039, 0.321]
C6	competitor, $L=0.025$	0.0655	[0.0622, 0.0696]	0.049	[0.033, 0.091]

Table 1: Final performance at 1000 evaluations. Identifiers match Fig. 3 ; the box defines the ablation.

Panel of Figure 4 — Figure 4: Left: Currents and position combined optimisation plots: (a) best $c_{Z}$ -so-far and (b) cumulative feasibility, vs our competitor method (C1-C8) . O1: joint with box trust region, O2: without. Right: (c) An illustration of the position bounds, (d) our optimal feasible separatrices from the combined optimisation and (e) from our currents-only optimisations.

Panel of Figure 5 — Figure 5: Left: HV plots from currents-only MOBO and currents and positions MOBO. Right: The corresponding connection length and total flux expansion pareto plots. Red are the joint MOBO results, blue the currents-only. The ‘x’ is our baseline equilibrium and the HV anchor.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

		currents	joint
Problem
	dimension $d$	$10$	$30$
	tolerance $\tau$ [cm]	$0.5$
	leg truncation margin [m]	$0.1$
	shot / index	$53349$ / $120$
	current bounds [A]	PX $\pm 6000$ , P4/P5 $\pm 14400$ , D1–D3 $\pm 8000$ ,

Appendix

Table 2: Experimental settings for the currents-only (

d=10

) and joint currents-and-positions (

d=30

) problems.

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Constraint-Bound Agnostic Bayesian Optimization: One Model for All Thresholds

Jin Wang, Xi Lin, Handing Wang

School of Artificial Intelligence, Xidian University, China · School of Mathematics and Statistics, Xi’an Jiaotong University, China

Bayesian Optimization Constrained Optimization

Open paper AskPDF arXiv

Expensive constrained optimization problems in real-world industry design often involve constraint thresholds that are difficult to determine in advance. Engineers may need to adjust constraint thresholds to explore different feasibility-performance trade-offs, requiring solutions under a wide range of threshold settings. However, existing constrained Bayesian optimization methods treat each threshold configuration independently, leading to repeated optimization and failing to exploit the shared relationship among continuously varying thresholds. To address this challenge, we propose constraint-bound agnostic Bayesian optimization (CBA-BO), a learning-based framework that learns a parametric constraint model mapping thresholds to optimal solutions. Once learned, CBA-BO directly predicts solutions for arbitrary unseen threshold configurations without additional optimization, with a one-step Bayesian optimization refinement further improving solution quality. Experiments on benchmark and engineering problems demonstrate that CBA-BO learns a transferable threshold-solution mapping, enabling efficient prediction and optimization for arbitrary threshold queries. An intent-guided constraint-bound recommendation mechanism is further developed to improve objective performance while satisfying user-specified constraint preferences.

Abstract

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