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
Figure 1: Left: (a) uniform samples across coil-current design variables 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.
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.
Figure 3: Currents-only ( d=10 ), 1000 evaluations. Median (bold), IQR (band) and every individual seed (faint). (a),(c) y-axis is best feasible cZ ; the dotted line is the baseline cZ=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 cZ
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.
Figure 4: Left: Currents and position combined optimisation plots: (a) best cZ -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.
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 τ [cm]
0.5
leg truncation margin [m]
0.1
shot / index
53349 / 120
current bounds [A]
PX ±6000 , P4/P5 ±14400 , D1–D3 ±8000 ,
Appendix
Table 2: Experimental settings for the currents-only ( d=10 ) and joint currents-and-positions ( d=30 ) problems.
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.
Jin Wang, Xi Lin, Handing Wang
School of Artificial Intelligence, Xidian University, China · School of Mathematics and Statistics, Xi’an Jiaotong University, China
Many design tasks can be cast as black-box function optimization, enabling use of Bayesian optimization to find an ideal design with minimal number of trials. However, often we do not actually need the optimum but instead a sufficiently good solution is enough, for instance a material that is durable enough for its intended use. In most cases there are multiple satisfactory solutions, forming a superlevel set of the function, raising a key question of which one to prefer. We answer this by explaining why robustness to input perturbations that may occur when the solution is deployed is a good criterion and by introduce a Bayesian optimization method that efficiently finds satisficing solutions that are robust to maximally large perturbations. In contrast to previous works, we assume the inputs can be accurately controlled during optimization, but will be perturbed after the deployment.
Samuli Kinnunen, Petrus Mikkola, Antti Niskanen +1
Department of Computer Science, University of Helsinki, Finland · ASM International N.V.
Bayesian Optimisation (BO) under unknown constraints is particularly challenging when feasible regions are small. In such settings, existing methods that typically rely solely on evaluations of the true objective and constraints struggle to efficiently explore the design space. However, many real-world applications offer auxiliary data sources (e.g. surrogate models or simplified simulations) that can support early exploration. Despite this potential, their integration into constrained BO remains largely unexplored. We propose a general multi-source framework that extends constrained Max-value Entropy Search, capturing inter-source correlation while balancing evaluation cost and information gain. Experiments on both synthetic and physics-based benchmarks show that our method efficiently identifies feasible and optimal solutions, even when auxiliary data are only weakly correlated. The proposed approach consistently outperforms existing methods, particularly in early-stage exploration.
Hauke Maathuis, Roeland De Breuker, Saullo Castro +1