A FEM-Based Surrogate Modelling and Optimization Framework for Physics-Constrained Electromagnetic Coil Design

Yucheng Liu

cs.LGAug 19, 2026

A FEM-Based Surrogate Modelling and Optimization Framework for Physics-Constrained Electromagnetic Coil Design

Authors: Yucheng Liu

Organizations: Department of Engineering Science University of Oxford Oxford, United Kingdom

Abstract

This work evaluates surrogate-assisted optimization of a seven-parameter current-excited coil--core benchmark subject to geometric, manufacturing, and separate core and copper mass constraints. A Python--MPh--COMSOL workflow couples a two-dimensional axisymmetric finite-element method (FEM) model to a Matern 5/2 Gaussian-process (GP) probabilistic surrogate. Here, physics-constrained denotes a design problem evaluated by a governing-equation FEM model and restricted by explicit physical, geometric, manufacturing, and material-allocation constraints; it does not denote a physics-informed GP architecture. Sequential Bayesian optimization (BO) ranks candidates using expected improvement (EI), and every reported incumbent is verified by FEM. Five paired runs show that optimizer ranking depends on the available FEM-evaluation budget: EI--BO improves rapidly at small continuation budgets, COBYLA is stronger at the earliest checkpoint, and BOBYQA attains the highest mean terminal response. A retrospective finite-pool study further finds no robust endpoint advantage of EI over posterior-mean ranking on this smooth response surface. The broader result is that early progress, terminal response, information use, and wall-clock cost can favor different methods in simulation-driven design. A selected-design check at a common total current preserves the observed BOBYQA--COBYLA--EI-BO ordering. The conclusions nevertheless remain conditional on this axisymmetric benchmark and do not establish a fixed-current optimum, fixed-power performance, or electrical-efficiency superiority.

Figures & tables

Study	FEM/EM	GP uncertainty	Sequential BO	Constraints	Repeated	Distinguishing scope
Li et al. [ 11 ]	Yes	No	No	Geometry	No	RF-coil $B_{1}$ homogeneity evaluated with field histograms.
Shi and Ludwig [ 12 ]	Yes	No	No	Application-specific	No	MRI gradient-coil optimization coupled directly to FEM.
Petkovska et al. [ 13 ]	Yes	No	No	Application-specific	No	FEM-coupled genetic optimization of electromagnetic-device designs.
Satonkar et al. [ 14 ]	Yes	No	No	Bounded DOE	No	Two-dimensional FEM with deterministic response-surface methodology.
This benchmark study	Yes	Yes	Yes	Geometry and mass	Five EI seeds	Held-out calibration, observed-FEM accounting, and a paired policy ablation.

TABLE I: Illustrative positioning against selected verified electromagnetic-design studies. The table is not an exhaustive survey. “Repeated” denotes repeated optimization runs, not repeated solver calls.

Panel of Figure 2 — Fig. 1: Seven-stage workflow used in the reported experiment.

Symbol	Description	Unit	Feasible range
$c_{1}$	Aperture radius	mm	$[0.5,\,5]$
$r_{1}$	Lower core radius	mm	$[10,\,20]$
$t_{1}$	Radial extension of upper core	mm	$[0.5,\,10]$
$l_{1}$	Lower core height	mm	$[2.5,\,20]$ , with $l_{1}\geq h_{1}+2$
$l_{2}$	Upper core height	mm	$[0.5,\,20]$
$w_{1}$	Coil radial width	mm	$[0.5,\,5]$

TABLE II: Independent design variables defining the parameterized benchmark geometry.

Symbol	Description	Unit	Value / Assumption
$\rho_{\mathrm{Cu}}$	Copper density	kg m -3	8940
$\sigma_{\mathrm{Cu}}$	Coil electrical conductivity	S m -1	$6.0\times 10^{7}$
$J_{\mathrm{apply}}$	Applied current-density parameter	A m -2	$1.0\times 10^{6}$
$\rho_{\mathrm{core}}$	Core density	kg m -3	7000
$B_{\mathrm{sat}}$	High Flux saturation flux density	T	1.5 [ 18 ]
Grade	Manufacturer nominal permeability grade	–	125 [ 18 ]

TABLE III: Fixed physical and material parameters for the regenerated benchmark.

Symbol	Description	Unit	Expression
$r_{2}$	Upper core radius	mm	$r_{1}+t_{1}$
$r_{0}$	Coil outer radius	mm	$r_{1}+w_{1}$
$M_{\mathrm{core}}$	Core mass	kg	$\rho_{\mathrm{core}}V_{\mathrm{core}}$
$M_{\mathrm{coil}}$	Coil mass	kg	$\rho_{\mathrm{Cu}}V_{\mathrm{coil}}$
$M_{\mathrm{total}}$	Total mass	kg	$M_{\mathrm{core}}+M_{\mathrm{coil}}$
$I_{\mathrm{coil}}$	Applied coil current	A	$J_{\mathrm{apply}}w_{1}h_{1}$

TABLE IV: Derived geometric and screening quantities.

Panel of Figure 6 — Fig. 2: Meridional schematic of the axisymmetric benchmark parameterization (not to scale). The dashed copper-coil domain carries the current excitation and is included in the geometric and mass constraints.

Panel of Figure 7 — Fig. 3: Data flow of the proposed surrogate-modelling framework. The bounded physical design space is represented using dimensionless coordinates in the unit hypercube, where Latin hypercube sampling is performed. Each normalized sample is mapped back to its corresponding physical geometry for constraint screening and FEM reference-model evaluation. The normalized coordinates and their associated FEM responses are subsequently paired to construct the Gaussian-process dataset.

Panel of Figure 8 — Fig. 4: Representative seed-42 learning curves: (a) $R^{2}$ , (b) RMSE, (c) MAE, (d) empirical $\pm 1\sigma$ coverage, and (e) empirical $\pm 2\sigma$ coverage.

$N_{\mathrm{train}}$	$R^{2}$	RMSE	MAE	$C_{1\sigma}$	$C_{2\sigma}$
25	$0.9872\pm 0.0041$	$20.32\pm 2.92$	$14.07\pm 1.63$	$0.516\pm 0.097$	$0.776\pm 0.085$
50	$0.9973\pm 0.0013$	$9.14\pm 2.17$	$6.83\pm 1.62$	$0.580\pm 0.110$	$0.856\pm 0.071$
75	$0.9985\pm 0.0008$	$6.77\pm 1.86$	$4.77\pm 1.14$	$0.572\pm 0.117$	$0.868\pm 0.061$
100	$0.9991\pm 0.0004$	$5.33\pm 1.12$	$3.61\pm 0.53$	$0.668\pm 0.046$	$0.920\pm 0.035$
125	$0.9992\pm 0.0004$	$4.96\pm 1.30$	$3.18\pm 0.75$	$0.720\pm 0.075$	$0.908\pm 0.030$
150	$0.9994\pm 0.0003$	$4.38\pm 1.39$	$2.78\pm 0.77$	$0.712\pm 0.083$	$0.908\pm 0.058$

TABLE V: Learning-curve metrics across five train–validation splits, reported as mean

\pm

sample SD. RMSE and MAE are in

\mu\mathrm{T}

Metric	Value
$R^{2}$	$0.9993$
RMSE	$4.680~\mu\mathrm{T}$
MAE	$2.235~\mu\mathrm{T}$
$C_{1\sigma}$	$0.76$
$C_{2\sigma}$	$0.90$
Standardized-residual mean	$-0.449$

TABLE VI: Held-out performance of the 200-sample seed-42 GP used for the detailed diagnostic plots.

Variable	Length scale	Rank
$h_{1}$	$21.10\pm 3.36$	1
$w_{1}$	$22.56\pm 4.14$	2
$t_{1}$	$28.74\pm 3.35$	3
$l_{1}$	$28.84\pm 3.18$	4
$l_{2}$	$35.29\pm 6.24$	5
$r_{1}$	$36.58\pm 3.82$	6

TABLE VII: ARD length scales from the five 200-sample fits. Values are mean

\pm

sample SD on standardized input coordinates; lower values indicate stronger surrogate-implied variation, not causal importance.

Panel of Figure 12 — Fig. 5: Surrogate diagnostics: (a) seed-42 parity with $\pm 1\sigma$ error bars; and five-split 200-sample diagnostics comprising (b) the reliability curve with a cross-split $\pm$ sample-SD band (not a confidence band), (c) absolute error versus predictive standard deviation, and (d) a pooled standardized-residual Q–Q plot.

Fig. 6: Best FEM-observed response versus cumulative accounted FEM observations over five paired seeds. Every trajectory includes the same seed-specific 25-point initial set. Bands are pointwise 95% Student- $t$ intervals, not simultaneous confidence bands.

Policy	Endpoint (mT)	Improvement (mT)
Random	$0.713\pm 0.055$	$0.039\pm 0.053$
scikit-learn posterior mean	$0.738\pm 0.000$	$0.063\pm 0.097$
scikit-learn latent EI	$0.738\pm 0.000$	$0.063\pm 0.097$
BoTorch posterior mean	$0.738\pm 0.000$	$0.063\pm 0.097$
BoTorch logarithmic EI	$0.738\pm 0.000$	$0.063\pm 0.097$

TABLE VIII: Paired retrospective 25+50 policy ablation on the finite FEM pool. Values are mean

\pm

sample SD over five shared initializations.

Method	Runs	Best response, mean $\pm$ SD (mT)	Min–max (mT)
BOBYQA	5	$0.943\pm 0.027$	$0.910$ – $0.974$
COBYLA	5	$0.940\pm 0.021$	$0.918$ – $0.965$
EI-BO	5	$0.909\pm 0.012$	$0.890$ – $0.920$
Nelder–Mead	5	$0.743\pm 0.062$	$0.636$ – $0.780$
EGO	5	$0.724\pm 0.033$	$0.665$ – $0.740$

TABLE IX: Terminal FEM-observed responses. Values are mean

\pm

sample SD and range over five paired seeds. COBYLA terminal values include native early stopping.

Method	$n=25$	$n=30$	$n=40$	$n=50$
EI-BO	0.675	0.883	0.890	0.893
BOBYQA	0.675	0.675	0.685	0.830
COBYLA	0.675	0.938	–	–
EGO	0.675	0.675	0.680	0.680
Nelder–Mead	0.675	0.680	0.696	0.713

TABLE X: Mean best-observed response at selected cumulative FEM budgets (mT), over five paired seeds. A dash indicates that fewer than five COBYLA runs remained active at the checkpoint.

Method	Mean accounted FEM calls	End-to-end time (s)
EI-BO	75	$178.3\pm 28.3$
BOBYQA	75	$116.4\pm 1.8$
COBYLA	36	$91.6\pm 27.7$
EGO	75	$109.6\pm 3.8$
Nelder–Mead	75	$122.0\pm 5.1$

TABLE XI: End-to-end costs under the reported implementations. Values are means over five seeds and include the common estimated 25-point initial-FEM cost.

Quantity	EI-BO (seed 45)	BOBYQA (seed 45)	COBYLA (seed 42)
$c_{1}$	1.2371	0.5000	3.0313
$r_{1}$	11.8847	10.0921	10.0000
$t_{1}$	9.3159	10.0000	10.0000
$l_{1}$	7.7991	7.0412	7.0000
$l_{2}$	14.8635	19.7206	18.0930
$w_{1}$	4.9086	5.0000	5.0000

TABLE XII: Best FEM-observed designs from EI-BO, BOBYQA, and COBYLA. Dimensions are in mm, masses in g, current in A, and response in mT.

Fig. 7: Fresh-solve magnetic-flux-density maps for the selected EI-BO, BOBYQA, and COBYLA geometries at the production current density. The red segment marks the meridional ROI whose revolution defines the averaging surface. A common color scale is used across panels.

Fig. 8: Radial magnetic-flux-density profiles on the ROI line (z=-10) mm for the three selected designs. The plotted curves are interpolated from the fresh FEM solution nodes; the reported scalar objective remains COMSOL’s axisymmetrically weighted boundary average.

Fig. 9: Manufacturer-sourced High Flux 125 input (B)–(H) curve and the core-domain mean-to-maximum (B) intervals from fresh solves of the selected geometries. Each interval is a field-magnitude range, not a paired (B(H)) trajectory.

Design	Production $I$ (A)	Production ROI (mT)	Common- $I$ ROI (mT)	Core mean $B$ (mT)	Core max $B$ (mT)	Common- $I$ rank
EI-BO	24.234	0.91952	0.91952	2.858	18.981	3
BOBYQA	25.000	0.97361	0.94378	2.615	63.424	1
COBYLA	25.000	0.96515	0.93558	2.816	19.444	2

TABLE XIII: Selected-geometry physical diagnostics. The common-current case uses 24.234 A, the minimum production current among the three designs; no re-optimization is performed.

Check	Setting	$B_{\mathrm{ROI}}$ (T)	Relative change
$l_{1}$ variation	7.7991 mm (optimized)	0.00091952	reference
	6.9370 mm	0.00093006	$+1.147\%$
	7.3681 mm	0.00092518	$+0.616\%$
	7.2991 mm	0.00092584	$+0.688\%$
	8.2991 mm	0.00091372	$-0.631\%$
Automatic mesh	hauto =3, 2574 elements	0.00091952	$-0.069\%$ vs. finest

TABLE XIV: Local sensitivity checks around the seed-45 EI-BO design. The

l_{1}

rows use the production mesh and 200-mm domain; mesh and domain rows retain the optimized geometry.

Design	Setting	Elements	$B_{\mathrm{ROI}}$ (T)	Relative change
BOBYQA	hauto =3, 200 mm	6342	0.00097361	$-0.0036\%$ vs. finest
	hauto =2, 200 mm	12166	0.00097384	$+0.0206\%$ vs. finest
	hauto =1, 200 mm	36134	0.00097364	reference
	hauto =3, 300 mm	6670	0.00097540	$-0.0556\%$ vs. largest
	hauto =3, 400 mm	6450	0.00097594	reference
COBYLA	hauto =3, 200 mm	2149	0.00096515	$+0.5478\%$ vs. finest

TABLE XV: Mesh and outer-domain checks at the selected BOBYQA and COBYLA geometries. Mesh changes are relative to hauto =1 at 200 mm; domain changes are relative to 400 mm at hauto =3.

Surrogate-Assisted Optimization

Surrogate-assisted optimization uses learned approximations to find strong solutions when experiments or simulations are costly.

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Surrogate Modeling Bayesian Optimization

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Bayesian Optimization (BO) is widely adopted for data-efficient optimization in scientific and engineering applications, yet its computational cost is rarely evaluated alongside optimization performance. Here we present a systematic, compute-aware study of BO that evaluates surrogate models along two axes: optimization quality and computational frugality. Across eight benchmark functions and nine real-world datasets spanning materials science, mechanics, robotics, chemistry, and machine learning, we benchmark four surrogate models: Gaussian Processes, Random Forests, NGBoost, and Bayesian Adaptive Spline Surfaces. We show that Gaussian Process-based BO consistently incurs the highest time and memory overhead without delivering superior optimization or sample efficiency. In contrast, scalable alternatives achieve equal or better performance at a fraction of the computational cost. Motivated by these findings, we introduce a surrogate-recommendation framework that predicts the most suitable BO surrogate from inexpensive dataset characteristics. Together, these results establish FruBO as a reproducible, compute-aware baseline for Bayesian Optimization and provide practical guidance for surrogate selection under limited computational and experimental budgets.

Abstract

Surrogate-Assisted Optimization

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