Bayesian optimization with kernel ensembles and disagreement-based acquisition for source localization and acoustic inversion

Heng Zhang; Haotian Xiang; Florian Meyer; Qin Lu

cs.LGSep 13, 2026

Bayesian optimization with kernel ensembles and disagreement-based acquisition for source localization and acoustic inversion

Authors: Heng Zhang, Haotian Xiang, Florian Meyer, Qin Lu

Organizations: School of Electrical and Computer Engineering, University of Georgia, Athens, GA, USA · Scripps Institution of Oceanography, University of California, San Diego, CA, USA

Abstract

Joint source localization and geoacoustic inversion requires optimizing an objective built from an expensive normal mode propagation model. Bayesian optimization (BO) with a Gaussian process (GP) surrogate can obtain accurate parameter estimates within a limited number of forward model evaluations, but its performance depends on the choice of kernel family. With few observations in a seven-dimensional search space, no single kernel can be expected to perform consistently well across individual inversions. To reduce this dependence, we use a weighted ensemble of GPs with different kernel families, allowing the surrogate to adapt to the observed objective without committing to one kernel in advance. The ensemble is combined with an optimum-conditioned acquisition function that determines where the expensive objective should be evaluated next. Experiments on simulated and measured SWellEx-96 data show that the resulting method achieves the lowest mean final objective among the considered BO strategies and reduces parameter estimation error on most coordinates. Ablation results further show that the ensemble provides robustness to kernel choice, while the acquisition function accounts for most of the optimization gain.

Figures & tables

Strategy	Data	RBF	Matérn- $3/2$	Matérn- $5/2$
BO-LogEI	Simulated	42%	33%	25%
BO-LogEI	Measured	49%	20%	31%
Local BO $q{=}1$	Simulated	32%	29%	39%
Local BO $q{=}1$	Measured	38%	22%	40%
Local BO $q{=}4$	Simulated	42%	18%	40%
Local BO $q{=}4$	Measured	44%	21%	35%

Table 1: Percentage of runs on which each kernel family gives the lowest normalized parameter error. Each run is repeated with all three families from an identical warm-up design, so the three are directly paired; 100 runs per row. No family is best on all the BO approaches.

Figure 1: Lowest observed objective $\widehat{\phi}_{t}$ against trial index for (a) simulated and (b) experimental data over 100 runs. Solid lines are the mean over runs, and dash-dot lines of the same colour the 10th and 90th percentile at that trial. The dots at the right edge show the distribution of the 100 final values. Curves separate only after the $T_{\mathrm{init}}=32$ warm-up trials (dashed vertical line), which are shared by all strategies at a given seed.

Symbol	Parameter	Bounds
$r_{\mathrm{src}}$ [km]	Source range	$[0.82,\ 1.32]$
$z_{\mathrm{src}}$ [m]	Source depth	$[60,\ 80]$
$\tau$ [ ∘ ]	Array tilt	$[-3,\ 3]$
$h_{w}$ [m]	Water depth	$[212,\ 222]$
$h_{s}$ [m]	Sediment thickness	$[15,\ 25]$
$c_{s,t}$ [m/s]	Sediment top speed	$[1560,\ 1580]$

Table 2: Search space

\Theta\subset\mathbb{R}^{D}

in ( 3 ),

D=7

Strategy	${\phi}^{*}$	$r_{\mathrm{src}}$ [km]	$z_{\mathrm{src}}$ [m]	$\tau$ [ ∘ ]	$h_{w}$ [m]	$h_{s}$ [m]	$c_{s,t}$ [m/s]	$c_{s,b}$ [m/s]
Best value	0 / 0.076	1.07 / 1.072	71.11 / 71.53	2 / 2.068	217 / 219.6	23 / 20.07	1572.3 / 1567	1593 / 1590
Random	0.0933 / 0.1983	0.028 / 0.035	2.04 / 1.96	0.392 / 0.399	2.63 / 3.00	3.15 / 2.53	5.24 / 5.62	6.47 / 5.96
BO-LogEI	0.0053 / 0.0813	0.018 / 0.012	0.81 / 0.27	0.195 / 0.041	1.98 / 1.42	1.81 / 1.73	6.64 / 7.55	7.03 / 6.24
Local BO $q{=}1$	0.0123 / 0.0942	0.023 / 0.019	1.03 / 0.62	0.178 / 0.098	2.55 / 1.95	1.85 / 2.39	6.73 / 8.10	7.16 / 8.60
Local BO $q{=}4$	0.0097 / 0.0902	0.023 / 0.015	1.05 / 0.62	0.168 / 0.101	2.30 / 1.22	1.53 / 1.39	5.65 / 6.25	5.92 / 5.75
KEDA	0.0017 / 0.0805	0.015 / 0.009	0.62 / 0.27	0.153 / 0.068	1.56 / 0.96	1.35 / 1.04	6.14 / 5.58	7.58 / 4.36

Table 3: Mean final objective and MAE of the seven estimated parameters over 100 runs, simulated / measured data. All baselines are our own reproduction under a common warm-up design. Best value of each column and track in bold.

Figure 2: Distribution of the 100 final KEDA estimates for (a)–(g) simulated and (h)–(n) measured data, 40 bins over the search bounds of each coordinate. The dashed line and marker give the true value.

Figure 3: KEDA with each single kernel of $\mathcal{K}$ and with the full ensemble, on (a) simulated and (b) experimental data, over 100 seeds.

Figure 4: The same four surrogates under log expected improvement instead of the disagreement acquisition.

Figure 5: KEDA against each baseline at the kernel minimising that baseline’s own mean final objective; the legend names the kernel used in each panel.

Figure 6: Kernel weights $w_{t}^{r}$ of ( 6 ) against trial $t$ on (a) simulated and (b) measured data, over 100 seeds. Solid lines give the mean over seeds and the shaded bands the interquartile range. The dotted line marks the uniform weight $1/R$ and the dashed line the floor $\epsilon/R$ . Weights exist only after the warm-up, so the curves begin at $t=T_{\mathrm{init}}+1$ .

Figure 7: Distribution over the 100 seeds of the final weight $w_{T}^{r}$ of each kernel in $\mathcal{K}$ , on (a) simulated and (b) measured data. The dashed line marks the floor $\epsilon/R$ ; bars exceeding the vertical range are labelled with their count, and the median of each panel is given at its top right.

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Boundary Variance Inflation Causes Acquisition Bias in Gaussian Processes

Maria Bånkestad, Sanna Jarl, Jens Sjölund

RISE Research Institutes of Sweden, Stockholm, Sweden · Uppsala University, Uppsala, Sweden

Bayesian Optimization Gaussian Processes

Open paper AskPDF arXiv

Gaussian processes with stationary kernels on bounded domains exhibit inflated posterior variance near the boundary. Despite being a long-recognized artifact in geostatistics and a source of over-exploration in Bayesian optimization, the causes and effects of boundary-induced acquisition bias are underexplored. We trace the root cause to a simple geometric mechanism: the truncation of the kernel correlation neighborhood at the domain boundary creates an observation-independent distortion that worsens with dimensionality. We show how this distortion manifests across three acquisition classes: variance maximization concentrates selections at the corners, whereas negative integrated posterior variance and expected predictive information gain move selections inward to axis-aligned interior shells. These patterns arise without reference to any objective function, meaning that acquisition behavior can be dominated by kernel geometry rather than the desired task-specific uncertainty. To quantify this, we introduce a function-free selection-profile diagnostic for arbitrary acquisitions, kernels, and bounded-domain geometries.

Abstract

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