MAST: A Multi-fidelity Augmented Surrogate model via Spatial Trust-weighting

Ahmed Mohamed Eisa Nasr; Ali Elham; Haris Moazam Sheikh

cs.LGFeb 24, 2026

MAST: A Multi-fidelity Augmented Surrogate model via Spatial Trust-weighting

Authors: Ahmed Mohamed Eisa Nasr, Ali Elham, Haris Moazam Sheikh

Organizations: Department of Aeronautics and Astronautics, University of Southampton, SO17 1BJ, Southampton, United Kingdom

Abstract

In engineering design and scientific computing, computational cost and predictive accuracy are intrinsically coupled. High-fidelity simulations provide accurate predictions but at substantial computational costs, while lower-fidelity approximations offer efficiency at the expense of accuracy. Multi-fidelity surrogate modelling addresses this trade-off by combining abundant low-fidelity data with sparse high-fidelity observations. However, existing methods rely on global correlation assumptions that can often fail in practice to capture how fidelity relationships vary across the input space, leading to poor performance, particularly under tight budget constraints. We introduce MAST, a method that blends corrected low-fidelity observations with high-fidelity predictions, trusting high-fidelity near observed samples and relying on corrected low-fidelity elsewhere. MAST achieves this through explicit discrepancy modelling and distance-based weighting with closed-form variance propagation, producing a single heteroscedastic Gaussian process. Across multi-fidelity synthetic benchmarks, MAST shows a marked improvement over the current state-of-the-art techniques. Crucially, MAST maintains robust performance across varying total budget and fidelity gaps, conditions under which competing methods exhibit significant degradation or unstable behaviour. More broadly, MAST provides a spatially adaptive framework for multi-fidelity Gaussian-process modelling, in which the contribution of low-fidelity information is governed by its proximity to high-fidelity calibration data, opening a new direction for more reliable surrogate construction under sparse and budget-constrained settings.

Figures & tables

	Normalised RMSE $\downarrow$									Normalised Mean PDF $\uparrow$
Test Function	MAST	AR1	Recur.	Reif.	BoTorch	NARGP	MF-DGP	MISO	MFBNN	MAST	AR1	Recur.	Reif.	BoTorch	NARGP	MF-DGP	MISO	MFBNN
Branin (2D)	1.19	2.06	6.03	2.24	0.95	1.67	1.86	2.23	1.18	1.28	0.63	0.62	1.37	1.43	0.89	0.91	1.35	1.05
Hartmann3 (3D)	0.69	0.29	1.29	0.41	0.29	0.48	0.92	0.42	1.07	1.83	3.97	0.72	2.52	2.48	2.45	0.94	2.35	1.08
Ackley (4D)	0.94	1.63	1.19	1.19	1.02	1.59	1.57	1.11	1.62	1.23	0.75	0.92	0.96	0.83	0.69	0.65	1.03	0.72
Park1 (4D)	0.97	1.15	4.99	2.46	1.06	1.24	3.00	2.31	1.25	1.46	1.38	0.48	0.37	0.72	1.35	0.26	0.42	0.87
Park2 (4D)	0.64	0.35	1.46	2.82	0.51	1.31	1.35	2.75	0.82	1.59	3.01	0.67	0.14	1.28	0.92	0.65	0.11	0.74

Table 1: Two-fidelity benchmarks: surrogate model performance comparison. Each cell shows the mean of the per-seed paired ratio (metric / HF-only baseline) across 25 seeds. RMSE values

<1

and Mean PDF

>1

indicate improvement. Best per row and metric in bold . Per-cell standard deviations of the same paired ratios are in Tables 11 (RMSE) and 13 (Mean PDF) (Appendix E.6 ).

	Normalised RMSE $\downarrow$									Normalised Mean PDF $\uparrow$
Test Function	MAST	AR1	Recur.	Reif.	BoTorch	NARGP	MF-DGP	MISO	MFBNN	MAST	AR1	Recur.	Reif.	BoTorch	NARGP	MF-DGP	MISO	MFBNN
Branin (2D)	1.06	2.04	1.96	1.83	0.64	1.91	1.25	1.78	1.37	1.24	0.68	0.63	1.12	1.55	0.43	0.95	1.20	0.91
Hartmann3 (3D)	0.76	0.55	1.06	9.01	0.30	0.97	0.97	0.58	1.42	1.60	2.24	0.85	1.61	2.28	1.27	0.63	1.85	0.80
Ackley (4D)	0.84	1.66	1.11	2.08	0.96	1.35	1.60	0.98	1.66	1.37	0.77	0.91	1.05	0.82	0.78	0.65	1.19	0.68
Park1 (4D)	1.15	1.17	2.10	3.68	1.23	1.57	2.20	1.84	2.80	1.21	1.09	0.59	0.54	0.61	0.92	0.23	0.53	0.43
Park2 (4D)	0.77	0.64	0.93	2.43	0.62	1.12	1.34	2.44	1.43	1.37	1.82	1.07	0.21	1.09	1.17	0.43	0.13	0.56

Table 2: Three-fidelity benchmarks: surrogate model performance comparison. Each cell shows the mean of the per-seed paired ratio (metric / HF-only baseline) across 25 seeds. RMSE values

<1

and Mean PDF

>1

indicate improvement. Best per row and metric in bold . Per-cell standard deviations of the same paired ratios are in Tables 12 (RMSE) and 14 (Mean PDF) (Appendix E.6 ).

Figure 1: Sensitivity to budget allocation. Top row: normalised RMSE; bottom row: normalised mean PDF on HOIP (3D), Borehole (8D), and Rosenbrock (20D), each divided by the cost-equivalent HF-only baseline (dotted line at 1.0). Full sweeps in Appendix E .

Figure 2: Sensitivity to total budget. Axes as in Figure 1 ; total budget rescales by

\{0.5,1,2,3\}\times\mathcal{B}

Figure 3: Sensitivity to fidelity discrepancy. Axes as in Figure 1 ; synthetic

d

of Eq. ( 13 ) varies

0.25{-}2.0

, HOIP LF-noise cost-ratio

\mathcal{C}

of Eq. ( 65 ) varies

1.0{-}50.0

Appendix figures & tables42 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: MAST pipeline. Stage 1: independent GPs at each fidelity. Stage 2:

N

-sphere trust region of radius

r_{i}=\sqrt{d_{\min}}

around each LF query; HF samples inside the sphere contribute to

W_{m}^{(i)}

. Stage 3: fused GP on Kumaraswamy-warped inputs with per-point noise bars from the propagated Stage 2 variance.

Figure 5: Stage 1 schematic. An independent Gaussian process is fitted at each fidelity. The predictive band at a training input equals the posterior latent variance plus the observation-noise variance, so the band pinches to near zero at every HF input (navy, low noise) but retains visible width at every LF input (teal, higher noise). The posterior reflects the noise on each observation rather than poor fit. The LF posterior is shifted down for visual separation only.

Figure 6: Stage 2 schematic. For a query low-fidelity point

\mathbf{x}_{i}^{(m)}

(filled teal), the trust region is an

N

-sphere of radius

r_{i}=\sqrt{d_{\min}}

centred on the query. High-fidelity samples inside the sphere contribute to the weight

W_{m}^{(i)}

with magnitude decaying in distance; high-fidelity samples outside are excluded. Dashed orange: trust-region boundary; orange arrows: distance-weighted contributions.

Figure 7: Stage 3 schematic. Left: per-dimension Kumaraswamy CDF

\Phi(x;\bm{a},\bm{b})

warps

[0,1]

onto itself (dotted: identity; navy: learned warp). Right: the Stage 3 GP is fitted on warped inputs with fixed per-point noise variances. High-fidelity observations (navy) have short noise bars; augmented low-fidelity points (teal) have longer bars reflecting the propagated Stage 2 variance. The fused posterior (navy line, shaded band) interpolates both sources.

Function class	$\sigma_{\mathrm{HF,spec}}$	$\sigma_{\mathrm{LF,spec}}$	Effective $\sigma_{\mathrm{LF}}$ at $c=1$
Ackley	0.02	0.13	0.15
All other functions	0.01	0.02	0.03

Appendix

Table 3: Per-function observation noise specifications. The low-fidelity effective noise at

d=1

\sigma_{\mathrm{HF,spec}}+\sigma_{\mathrm{LF,spec}}

, applied as a fraction of

|\mathbb{E}[f_{\mathrm{LF}}]|

per Eq. ( 42 ).

Function	Dim.	Domain	Discrepancy Type
Branin	2	$[-5,10]\times[0,15]$	Coefficient perturbation
Hartmann3	3	$[0,1]^{3}$	Parameter perturbation
Ackley	4	$[-5,5]^{4}$	Noise only
Park1	4	$[0,1]^{4}$	Multiplicative + polynomial
Park2	4	$[0,1]^{4}$	Linear scaling + offset
Hartmann6	6	$[0,1]^{6}$	Parameter perturbation

Appendix

Table 4: Summary of benchmark test functions in the active evaluation roster.

Test function	R1: $\sqrt{d_{\min}}$ (default)	R2: $d_{\min}$	R3: $d_{\min}^{1/3}$	R4: $10^{-6}$ (zero+)	R5: $\infty$ (global)
Branin (2D)	$1.192{\scriptstyle\pm 0.397}$	$\mathbf{1.163{\scriptstyle\pm 0.393}}$	$1.222{\scriptstyle\pm 0.419}$	$\mathbf{1.163{\scriptstyle\pm 0.393}}$	$1.334{\scriptstyle\pm 0.488}$
Hartmann-3 (3D)	$0.692{\scriptstyle\pm 0.188}$	$0.799{\scriptstyle\pm 0.221}$	$0.623{\scriptstyle\pm 0.168}$	$0.799{\scriptstyle\pm 0.221}$	$\mathbf{0.464{\scriptstyle\pm 0.125}}$
Ackley (4D)	$0.945{\scriptstyle\pm 0.256}$	$0.954{\scriptstyle\pm 0.276}$	$\mathbf{0.941{\scriptstyle\pm 0.247}}$	$0.954{\scriptstyle\pm 0.276}$	$0.945{\scriptstyle\pm 0.233}$
Park-1 (4D)	$0.970{\scriptstyle\pm 0.475}$	$0.978{\scriptstyle\pm 0.474}$	$\mathbf{0.963{\scriptstyle\pm 0.486}}$	$0.978{\scriptstyle\pm 0.474}$	$1.108{\scriptstyle\pm 0.498}$
Park-2 (4D)	$0.644{\scriptstyle\pm 0.177}$	$0.704{\scriptstyle\pm 0.198}$	$0.605{\scriptstyle\pm 0.161}$	$0.704{\scriptstyle\pm 0.198}$	$\mathbf{0.560{\scriptstyle\pm 0.133}}$
Hartmann-6 (6D)	$0.882{\scriptstyle\pm 0.193}$	$0.881{\scriptstyle\pm 0.188}$	$\mathbf{0.873{\scriptstyle\pm 0.180}}$	$0.881{\scriptstyle\pm 0.188}$	$0.874{\scriptstyle\pm 0.176}$

Appendix

Table 5: Trust-region radius function ablation — normalised RMSE per benchmark, mean

\pm

std across 25 seeds; lower is better. Best variant per function in bold.

Test function	R1: $\sqrt{d_{\min}}$ (default)	R2: $d_{\min}$	R3: $d_{\min}^{1/3}$	R4: $10^{-6}$ (zero+)	R5: $\infty$ (global)
Branin (2D)	$1.279{\scriptstyle\pm 0.427}$	$\mathbf{1.325{\scriptstyle\pm 0.480}}$	$1.284{\scriptstyle\pm 0.423}$	$\mathbf{1.325{\scriptstyle\pm 0.480}}$	$1.228{\scriptstyle\pm 0.438}$
Hartmann-3 (3D)	$1.826{\scriptstyle\pm 0.500}$	$1.663{\scriptstyle\pm 0.475}$	$1.996{\scriptstyle\pm 0.536}$	$1.663{\scriptstyle\pm 0.475}$	$\mathbf{2.496{\scriptstyle\pm 0.637}}$
Ackley (4D)	$1.229{\scriptstyle\pm 0.321}$	$\mathbf{1.241{\scriptstyle\pm 0.326}}$	$1.218{\scriptstyle\pm 0.314}$	$\mathbf{1.241{\scriptstyle\pm 0.326}}$	$1.166{\scriptstyle\pm 0.289}$
Park-1 (4D)	$1.460{\scriptstyle\pm 0.494}$	$1.445{\scriptstyle\pm 0.487}$	$\mathbf{1.464{\scriptstyle\pm 0.484}}$	$1.445{\scriptstyle\pm 0.487}$	$1.285{\scriptstyle\pm 0.387}$
Park-2 (4D)	$1.590{\scriptstyle\pm 0.284}$	$1.488{\scriptstyle\pm 0.264}$	$1.663{\scriptstyle\pm 0.294}$	$1.488{\scriptstyle\pm 0.264}$	$\mathbf{1.795{\scriptstyle\pm 0.338}}$
Hartmann-6 (6D)	$1.792{\scriptstyle\pm 0.523}$	$\mathbf{1.806{\scriptstyle\pm 0.523}}$	$1.791{\scriptstyle\pm 0.531}$	$\mathbf{1.806{\scriptstyle\pm 0.523}}$	$1.749{\scriptstyle\pm 0.534}$

Appendix

Table 6: Trust-region radius function ablation — normalised mean PDF per benchmark, mean

\pm

std across 25 seeds; higher is better. Best variant per function in bold.

Test function	W1: $1{-}d^{\alpha}$ (def.)	W2: $e^{-\alpha d}$	W3: $e^{-\alpha d^{2}}$	W4: $(1{+}d^{\alpha})^{-1}$	W5: triangular	W6: $1{-}d$	W7: $1{-}d^{2\alpha}$	W8: $e^{-d^{2}}$	W9: $h{=}1$
Branin (2D)	$1.192{\scriptstyle\pm 0.397}$	$1.206{\scriptstyle\pm 0.427}$	$1.284{\scriptstyle\pm 0.485}$	$\mathbf{1.112{\scriptstyle\pm 0.339}}$	$1.236{\scriptstyle\pm 0.417}$	$1.113{\scriptstyle\pm 0.344}$	$1.113{\scriptstyle\pm 0.344}$	$1.231{\scriptstyle\pm 0.444}$	$1.346{\scriptstyle\pm 0.524}$
Hartmann-3 (3D)	$0.692{\scriptstyle\pm 0.188}$	$1.321{\scriptstyle\pm 0.431}$	$1.417{\scriptstyle\pm 0.479}$	$1.066{\scriptstyle\pm 0.325}$	$\mathbf{0.576{\scriptstyle\pm 0.150}}$	$1.035{\scriptstyle\pm 0.308}$	$1.035{\scriptstyle\pm 0.308}$	$1.341{\scriptstyle\pm 0.437}$	$1.481{\scriptstyle\pm 0.524}$
Ackley (4D)	$0.945{\scriptstyle\pm 0.256}$	$1.204{\scriptstyle\pm 0.389}$	$1.271{\scriptstyle\pm 0.414}$	$1.026{\scriptstyle\pm 0.305}$	$\mathbf{0.931{\scriptstyle\pm 0.243}}$	$1.005{\scriptstyle\pm 0.304}$	$1.005{\scriptstyle\pm 0.304}$	$1.208{\scriptstyle\pm 0.398}$	$1.327{\scriptstyle\pm 0.422}$
Park-1 (4D)	$0.970{\scriptstyle\pm 0.475}$	$1.362{\scriptstyle\pm 0.570}$	$1.482{\scriptstyle\pm 0.613}$	$1.145{\scriptstyle\pm 0.506}$	$\mathbf{0.969{\scriptstyle\pm 0.483}}$	$1.060{\scriptstyle\pm 0.479}$	$1.059{\scriptstyle\pm 0.479}$	$1.353{\scriptstyle\pm 0.560}$	$1.614{\scriptstyle\pm 0.677}$
Park-2 (4D)	$0.644{\scriptstyle\pm 0.177}$	$1.285{\scriptstyle\pm 0.404}$	$1.408{\scriptstyle\pm 0.451}$	$1.020{\scriptstyle\pm 0.306}$	$\mathbf{0.591{\scriptstyle\pm 0.154}}$	$0.880{\scriptstyle\pm 0.249}$	$0.880{\scriptstyle\pm 0.249}$	$1.269{\scriptstyle\pm 0.391}$	$1.566{\scriptstyle\pm 0.522}$
Hartmann-6 (6D)	$\mathbf{0.882{\scriptstyle\pm 0.193}}$	$1.049{\scriptstyle\pm 0.154}$	$1.090{\scriptstyle\pm 0.160}$	$0.962{\scriptstyle\pm 0.156}$	$0.882{\scriptstyle\pm 0.183}$	$0.915{\scriptstyle\pm 0.169}$	$0.915{\scriptstyle\pm 0.169}$	$1.024{\scriptstyle\pm 0.153}$	$1.129{\scriptstyle\pm 0.156}$

Appendix

Table 7: Distance weight function ablation — normalised RMSE per benchmark, mean

\pm

std across 25 seeds; lower is better. Best variant per function in bold.

Test function	W1: $1{-}d^{\alpha}$ (def.)	W2: $e^{-\alpha d}$	W3: $e^{-\alpha d^{2}}$	W4: $(1{+}d^{\alpha})^{-1}$	W5: triangular	W6: $1{-}d$	W7: $1{-}d^{2\alpha}$	W8: $e^{-d^{2}}$	W9: $h{=}1$
Branin (2D)	$1.279{\scriptstyle\pm 0.427}$	$1.203{\scriptstyle\pm 0.471}$	$1.131{\scriptstyle\pm 0.458}$	$\mathbf{1.334{\scriptstyle\pm 0.458}}$	$1.245{\scriptstyle\pm 0.408}$	$1.287{\scriptstyle\pm 0.462}$	$1.287{\scriptstyle\pm 0.462}$	$1.167{\scriptstyle\pm 0.465}$	$1.089{\scriptstyle\pm 0.458}$
Hartmann-3 (3D)	$1.826{\scriptstyle\pm 0.500}$	$1.228{\scriptstyle\pm 0.418}$	$1.156{\scriptstyle\pm 0.418}$	$1.411{\scriptstyle\pm 0.428}$	$\mathbf{2.079{\scriptstyle\pm 0.539}}$	$1.413{\scriptstyle\pm 0.426}$	$1.413{\scriptstyle\pm 0.426}$	$1.206{\scriptstyle\pm 0.419}$	$1.124{\scriptstyle\pm 0.409}$
Ackley (4D)	$\mathbf{1.229{\scriptstyle\pm 0.321}}$	$1.051{\scriptstyle\pm 0.232}$	$0.996{\scriptstyle\pm 0.227}$	$1.189{\scriptstyle\pm 0.259}$	$1.224{\scriptstyle\pm 0.319}$	$1.214{\scriptstyle\pm 0.285}$	$1.214{\scriptstyle\pm 0.285}$	$1.048{\scriptstyle\pm 0.238}$	$0.951{\scriptstyle\pm 0.209}$
Park-1 (4D)	$\mathbf{1.460{\scriptstyle\pm 0.494}}$	$1.086{\scriptstyle\pm 0.359}$	$1.015{\scriptstyle\pm 0.349}$	$1.259{\scriptstyle\pm 0.399}$	$1.457{\scriptstyle\pm 0.488}$	$1.324{\scriptstyle\pm 0.421}$	$1.324{\scriptstyle\pm 0.421}$	$1.089{\scriptstyle\pm 0.361}$	$0.947{\scriptstyle\pm 0.346}$
Park-2 (4D)	$1.590{\scriptstyle\pm 0.284}$	$0.944{\scriptstyle\pm 0.180}$	$0.870{\scriptstyle\pm 0.168}$	$1.147{\scriptstyle\pm 0.204}$	$\mathbf{1.700{\scriptstyle\pm 0.304}}$	$1.263{\scriptstyle\pm 0.222}$	$1.263{\scriptstyle\pm 0.222}$	$0.947{\scriptstyle\pm 0.181}$	$0.797{\scriptstyle\pm 0.158}$
Hartmann-6 (6D)	$\mathbf{1.792{\scriptstyle\pm 0.523}}$	$1.732{\scriptstyle\pm 0.578}$	$1.692{\scriptstyle\pm 0.575}$	$1.757{\scriptstyle\pm 0.550}$	$1.789{\scriptstyle\pm 0.532}$	$1.790{\scriptstyle\pm 0.554}$	$1.791{\scriptstyle\pm 0.554}$	$1.712{\scriptstyle\pm 0.553}$	$1.602{\scriptstyle\pm 0.557}$

Appendix

Table 8: Distance weight function ablation — normalised mean PDF per benchmark, mean

\pm

std across 25 seeds; higher is better. Best variant per function in bold.

Test function	D1: Euclidean (default)	D2: Manhattan	D3: Chebyshev	D4: Mahalanobis (ARD)	D5: Norm. Euclidean	D6: Domain-relative
Branin (2D)	$1.192{\scriptstyle\pm 0.397}$	$1.228{\scriptstyle\pm 0.422}$	$1.184{\scriptstyle\pm 0.392}$	$1.342{\scriptstyle\pm 0.503}$	$\mathbf{1.162{\scriptstyle\pm 0.378}}$	$1.192{\scriptstyle\pm 0.397}$
Hartmann-3 (3D)	$0.692{\scriptstyle\pm 0.188}$	$0.589{\scriptstyle\pm 0.153}$	$0.745{\scriptstyle\pm 0.205}$	$\mathbf{0.430{\scriptstyle\pm 0.139}}$	$0.849{\scriptstyle\pm 0.253}$	$0.692{\scriptstyle\pm 0.188}$
Ackley (4D)	$0.945{\scriptstyle\pm 0.256}$	$\mathbf{0.936{\scriptstyle\pm 0.230}}$	$0.941{\scriptstyle\pm 0.266}$	$0.954{\scriptstyle\pm 0.251}$	$0.970{\scriptstyle\pm 0.279}$	$0.945{\scriptstyle\pm 0.256}$
Park-1 (4D)	$\mathbf{0.970{\scriptstyle\pm 0.475}}$	$0.989{\scriptstyle\pm 0.483}$	$0.982{\scriptstyle\pm 0.479}$	$0.977{\scriptstyle\pm 0.474}$	$1.014{\scriptstyle\pm 0.477}$	$\mathbf{0.970{\scriptstyle\pm 0.475}}$
Park-2 (4D)	$0.644{\scriptstyle\pm 0.177}$	$\mathbf{0.576{\scriptstyle\pm 0.141}}$	$0.710{\scriptstyle\pm 0.212}$	$0.667{\scriptstyle\pm 0.175}$	$0.796{\scriptstyle\pm 0.228}$	$0.644{\scriptstyle\pm 0.177}$
Hartmann-6 (6D)	$0.882{\scriptstyle\pm 0.193}$	$\mathbf{0.873{\scriptstyle\pm 0.177}}$	$0.887{\scriptstyle\pm 0.183}$	$0.888{\scriptstyle\pm 0.193}$	$0.909{\scriptstyle\pm 0.168}$	$0.882{\scriptstyle\pm 0.193}$

Appendix

Table 9: Distance-metric ablation — normalised RMSE per benchmark, mean

\pm

std across 25 seeds; lower is better. Best variant per function in bold. D6 (Domain-relative) reduces to D1 after Stage-1 input normalisation to

[0,1]^{D}

and is included as an implementation sanity check; its column matches D1 by construction.

Test function	D1: Euclidean (default)	D2: Manhattan	D3: Chebyshev	D4: Mahalanobis (ARD)	D5: Norm. Euclidean	D6: Domain-relative
Branin (2D)	$1.279{\scriptstyle\pm 0.427}$	$1.278{\scriptstyle\pm 0.434}$	$1.287{\scriptstyle\pm 0.424}$	$1.236{\scriptstyle\pm 0.433}$	$\mathbf{1.299{\scriptstyle\pm 0.432}}$	$1.279{\scriptstyle\pm 0.427}$
Hartmann-3 (3D)	$1.826{\scriptstyle\pm 0.500}$	$2.058{\scriptstyle\pm 0.562}$	$1.742{\scriptstyle\pm 0.483}$	$\mathbf{2.697{\scriptstyle\pm 0.695}}$	$1.614{\scriptstyle\pm 0.453}$	$1.826{\scriptstyle\pm 0.500}$
Ackley (4D)	$1.229{\scriptstyle\pm 0.321}$	$1.196{\scriptstyle\pm 0.282}$	$\mathbf{1.251{\scriptstyle\pm 0.317}}$	$1.156{\scriptstyle\pm 0.286}$	$1.235{\scriptstyle\pm 0.306}$	$1.229{\scriptstyle\pm 0.321}$
Park-1 (4D)	$1.460{\scriptstyle\pm 0.494}$	$1.428{\scriptstyle\pm 0.485}$	$1.452{\scriptstyle\pm 0.485}$	$\mathbf{1.462{\scriptstyle\pm 0.506}}$	$1.408{\scriptstyle\pm 0.460}$	$1.460{\scriptstyle\pm 0.494}$
Park-2 (4D)	$1.590{\scriptstyle\pm 0.284}$	$\mathbf{1.725{\scriptstyle\pm 0.318}}$	$1.499{\scriptstyle\pm 0.268}$	$1.548{\scriptstyle\pm 0.272}$	$1.373{\scriptstyle\pm 0.235}$	$1.590{\scriptstyle\pm 0.284}$
Hartmann-6 (6D)	$\mathbf{1.792{\scriptstyle\pm 0.523}}$	$1.741{\scriptstyle\pm 0.530}$	$1.785{\scriptstyle\pm 0.515}$	$1.731{\scriptstyle\pm 0.533}$	$1.771{\scriptstyle\pm 0.517}$	$\mathbf{1.792{\scriptstyle\pm 0.523}}$

Appendix

Table 10: Distance-metric ablation — normalised mean PDF per benchmark, mean

\pm

std across 25 seeds; higher is better. Best variant per function in bold. D6 (Domain-relative) reduces to D1 after Stage-1 input normalisation to

[0,1]^{D}

and is included as an implementation sanity check; its column matches D1 by construction.

Figure 8: Synthetic budget-allocation sweep, group 1: Branin (2D), Hartmann3 (3D), Ackley (4D). Top: normalised RMSE; bottom: normalised mean PDF.

Figure 9: Synthetic budget-allocation sweep, group 2: Park1 (4D), Park2 (4D), Hartmann6 (6D).

Figure 10: Synthetic budget-allocation sweep, group 3: Levy (7D), Borehole (8D), Rastrigin (15D).

Figure 11: Synthetic budget-allocation sweep, group 4: Rosenbrock (20D).

Figure 12: Real-world budget-allocation sweep: Concrete (8D), HOIP (3D), DrivAerNet (23D).

Figure 13: Real-world budget-allocation sweep: HOIP and DrivAerNet drawn from independent training pools at each fidelity, in contrast to the collocated pairing in Figure 12 .

Figure 14: Synthetic total-budget sweep, group 1: Branin, Hartmann3, Ackley.

Figure 15: Synthetic total-budget sweep, group 2: Park1, Park2, Hartmann6.

Figure 16: Synthetic total-budget sweep, group 3: Levy, Borehole, Rastrigin.

Figure 17: Synthetic total-budget sweep, group 4: Rosenbrock.

Figure 18: Real-world total-budget sweep: Concrete, HOIP, DrivAerNet.

Figure 19: Real-world total-budget sweep: HOIP and DrivAerNet drawn from independent pools.

Figure 20: Synthetic discrepancy sweep against

d

, group 1: Branin, Hartmann3, Ackley.

Figure 21: Synthetic discrepancy sweep against

d

, group 2: Park1, Park2, Hartmann6.

Figure 22: Synthetic discrepancy sweep against

d

, group 3: Levy, Borehole, Rastrigin.

Figure 23: Synthetic discrepancy sweep against

d

, group 4: Rosenbrock.

Figure 24: Same data as Figure 20 , with

N_{\mathrm{LF}}

on the x-axis instead of

d

. The two axes are linked by

N_{\mathrm{LF}}/N_{\mathrm{HF}}=10\,d

Figure 25: Same data as Figure 21 , plotted against

N_{\mathrm{LF}}

Figure 26: Same data as Figure 22 , plotted against

N_{\mathrm{LF}}

Figure 27: Same data as Figure 23 , plotted against

N_{\mathrm{LF}}

Figure 28: Real-world LF-noise-degradation sweep: Concrete, HOIP, DrivAerNet. The x-axis is the HF/LF cost ratio

\mathcal{C}

; the LF observations are corrupted with extra Gaussian noise per Eq. ( 65 ).

Figure 29: Real-world LF-noise-degradation sweep: HOIP and DrivAerNet drawn from independent pools.

Test Function	MAST	AR1	Recur.	Reif.	BoTorch	NARGP	MF-DGP	MISO	MFBNN
Branin (2D)	$1.19{\scriptstyle\pm 0.40}$	$2.06{\scriptstyle\pm 1.07}$	$6.03{\scriptstyle\pm 10.61}$	$2.24{\scriptstyle\pm 0.76}$	$\mathbf{0.95{\scriptstyle\pm 0.41}}$	$1.67{\scriptstyle\pm 0.80}$	$1.86{\scriptstyle\pm 1.65}$	$2.23{\scriptstyle\pm 0.76}$	$1.18{\scriptstyle\pm 0.35}$
Hartmann3 (3D)	$0.69{\scriptstyle\pm 0.19}$	$\mathbf{0.29{\scriptstyle\pm 0.09}}$	$1.29{\scriptstyle\pm 0.37}$	$0.41{\scriptstyle\pm 0.14}$	$0.29{\scriptstyle\pm 0.08}$	$0.48{\scriptstyle\pm 0.25}$	$0.92{\scriptstyle\pm 0.58}$	$0.42{\scriptstyle\pm 0.14}$	$1.07{\scriptstyle\pm 0.28}$
Ackley (4D)	$\mathbf{0.94{\scriptstyle\pm 0.26}}$	$1.63{\scriptstyle\pm 0.47}$	$1.19{\scriptstyle\pm 0.39}$	$1.19{\scriptstyle\pm 0.42}$	$1.02{\scriptstyle\pm 0.34}$	$1.59{\scriptstyle\pm 0.52}$	$1.57{\scriptstyle\pm 0.44}$	$1.11{\scriptstyle\pm 0.38}$	$1.62{\scriptstyle\pm 0.69}$
Park1 (4D)	$\mathbf{0.97{\scriptstyle\pm 0.48}}$	$1.15{\scriptstyle\pm 1.15}$	$4.99{\scriptstyle\pm 9.24}$	$2.46{\scriptstyle\pm 0.70}$	$1.06{\scriptstyle\pm 0.30}$	$1.24{\scriptstyle\pm 0.94}$	$3.00{\scriptstyle\pm 1.65}$	$2.31{\scriptstyle\pm 0.67}$	$1.25{\scriptstyle\pm 0.38}$
Park2 (4D)	$0.64{\scriptstyle\pm 0.18}$	$\mathbf{0.35{\scriptstyle\pm 0.11}}$	$1.46{\scriptstyle\pm 0.46}$	$2.82{\scriptstyle\pm 0.44}$	$0.51{\scriptstyle\pm 0.11}$	$1.31{\scriptstyle\pm 0.56}$	$1.35{\scriptstyle\pm 0.57}$	$2.75{\scriptstyle\pm 0.43}$	$0.82{\scriptstyle\pm 0.25}$
Hartmann6 (6D)	$0.88{\scriptstyle\pm 0.19}$	$0.67{\scriptstyle\pm 0.11}$	$1.00{\scriptstyle\pm 0.10}$	$0.84{\scriptstyle\pm 0.14}$	$\mathbf{0.64{\scriptstyle\pm 0.09}}$	$0.93{\scriptstyle\pm 0.21}$	$1.09{\scriptstyle\pm 0.15}$	$0.81{\scriptstyle\pm 0.13}$	$1.02{\scriptstyle\pm 0.18}$

Appendix

Table 11: Two-fidelity benchmarks: normalised RMSE (

\downarrow

, lower is better;

<1

beats HF-only baseline). Mean

\pm

std of the per-seed paired ratio across 25 seeds. Best per row in bold . Mean PDF counterpart: Table 13 ; combined-mean main-body view: Table 1 .

Test Function	MAST	AR1	Recur.	Reif.	BoTorch	NARGP	MF-DGP	MISO	MFBNN
Branin (2D)	$1.06{\scriptstyle\pm 0.29}$	$2.04{\scriptstyle\pm 1.32}$	$1.96{\scriptstyle\pm 0.63}$	$1.83{\scriptstyle\pm 0.59}$	$\mathbf{0.64{\scriptstyle\pm 0.23}}$	$1.91{\scriptstyle\pm 0.64}$	$1.25{\scriptstyle\pm 0.61}$	$1.78{\scriptstyle\pm 0.57}$	$1.37{\scriptstyle\pm 0.47}$
Hartmann3 (3D)	$0.76{\scriptstyle\pm 0.18}$	$0.55{\scriptstyle\pm 0.22}$	$1.06{\scriptstyle\pm 0.31}$	$9.01{\scriptstyle\pm 35.37}$	$\mathbf{0.30{\scriptstyle\pm 0.08}}$	$0.97{\scriptstyle\pm 0.42}$	$0.97{\scriptstyle\pm 0.35}$	$0.58{\scriptstyle\pm 0.16}$	$1.42{\scriptstyle\pm 0.39}$
Ackley (4D)	$\mathbf{0.84{\scriptstyle\pm 0.23}}$	$1.66{\scriptstyle\pm 0.37}$	$1.11{\scriptstyle\pm 0.39}$	$2.08{\scriptstyle\pm 1.68}$	$0.96{\scriptstyle\pm 0.29}$	$1.35{\scriptstyle\pm 0.33}$	$1.60{\scriptstyle\pm 0.52}$	$0.98{\scriptstyle\pm 0.30}$	$1.66{\scriptstyle\pm 0.60}$
Park1 (4D)	$\mathbf{1.15{\scriptstyle\pm 0.40}}$	$1.17{\scriptstyle\pm 0.49}$	$2.10{\scriptstyle\pm 1.27}$	$3.68{\scriptstyle\pm 4.96}$	$1.23{\scriptstyle\pm 0.42}$	$1.57{\scriptstyle\pm 1.36}$	$2.20{\scriptstyle\pm 0.92}$	$1.84{\scriptstyle\pm 0.54}$	$2.80{\scriptstyle\pm 1.18}$
Park2 (4D)	$0.77{\scriptstyle\pm 0.23}$	$0.64{\scriptstyle\pm 0.24}$	$0.93{\scriptstyle\pm 0.59}$	$2.43{\scriptstyle\pm 0.40}$	$\mathbf{0.62{\scriptstyle\pm 0.14}}$	$1.12{\scriptstyle\pm 0.52}$	$1.34{\scriptstyle\pm 0.49}$	$2.44{\scriptstyle\pm 0.37}$	$1.43{\scriptstyle\pm 0.42}$
Hartmann6 (6D)	$0.91{\scriptstyle\pm 0.19}$	$0.93{\scriptstyle\pm 0.27}$	$1.00{\scriptstyle\pm 0.18}$	$1.82{\scriptstyle\pm 1.99}$	$\mathbf{0.65{\scriptstyle\pm 0.10}}$	$1.19{\scriptstyle\pm 0.28}$	$1.22{\scriptstyle\pm 0.28}$	$0.87{\scriptstyle\pm 0.16}$	$1.17{\scriptstyle\pm 0.27}$

Appendix

Table 12: Three-fidelity benchmarks: normalised RMSE (

\downarrow

, lower is better;

<1

beats HF-only baseline). Mean

\pm

std of the per-seed paired ratio across 25 seeds. Best per row in bold . Mean PDF counterpart: Table 14 ; combined-mean main-body view: Table 2 .

Test Function	MAST	AR1	Recur.	Reif.	BoTorch	NARGP	MF-DGP	MISO	MFBNN
Branin (2D)	$1.28{\scriptstyle\pm 0.43}$	$0.63{\scriptstyle\pm 0.75}$	$0.62{\scriptstyle\pm 0.47}$	$1.37{\scriptstyle\pm 0.44}$	$\mathbf{1.43{\scriptstyle\pm 0.46}}$	$0.89{\scriptstyle\pm 0.68}$	$0.91{\scriptstyle\pm 0.46}$	$1.35{\scriptstyle\pm 0.44}$	$1.05{\scriptstyle\pm 0.40}$
Hartmann3 (3D)	$1.83{\scriptstyle\pm 0.50}$	$\mathbf{3.97{\scriptstyle\pm 0.74}}$	$0.72{\scriptstyle\pm 0.18}$	$2.52{\scriptstyle\pm 0.63}$	$2.48{\scriptstyle\pm 0.51}$	$2.45{\scriptstyle\pm 0.77}$	$0.94{\scriptstyle\pm 0.28}$	$2.35{\scriptstyle\pm 0.56}$	$1.08{\scriptstyle\pm 0.34}$
Ackley (4D)	$\mathbf{1.23{\scriptstyle\pm 0.32}}$	$0.75{\scriptstyle\pm 0.20}$	$0.92{\scriptstyle\pm 0.23}$	$0.96{\scriptstyle\pm 0.23}$	$0.83{\scriptstyle\pm 0.21}$	$0.69{\scriptstyle\pm 0.16}$	$0.65{\scriptstyle\pm 0.13}$	$1.03{\scriptstyle\pm 0.24}$	$0.72{\scriptstyle\pm 0.24}$
Park1 (4D)	$\mathbf{1.46{\scriptstyle\pm 0.49}}$	$1.38{\scriptstyle\pm 0.64}$	$0.48{\scriptstyle\pm 0.32}$	$0.37{\scriptstyle\pm 0.06}$	$0.72{\scriptstyle\pm 0.08}$	$1.35{\scriptstyle\pm 0.67}$	$0.26{\scriptstyle\pm 0.07}$	$0.42{\scriptstyle\pm 0.06}$	$0.87{\scriptstyle\pm 0.15}$
Park2 (4D)	$1.59{\scriptstyle\pm 0.28}$	$\mathbf{3.01{\scriptstyle\pm 0.69}}$	$0.67{\scriptstyle\pm 0.23}$	$0.14{\scriptstyle\pm 0.06}$	$1.28{\scriptstyle\pm 0.15}$	$0.92{\scriptstyle\pm 0.21}$	$0.65{\scriptstyle\pm 0.19}$	$0.11{\scriptstyle\pm 0.06}$	$0.74{\scriptstyle\pm 0.09}$
Hartmann6 (6D)	$\mathbf{1.80{\scriptstyle\pm 0.52}}$	$1.50{\scriptstyle\pm 0.46}$	$1.06{\scriptstyle\pm 0.41}$	$1.42{\scriptstyle\pm 0.40}$	$1.17{\scriptstyle\pm 0.29}$	$1.30{\scriptstyle\pm 0.59}$	$0.89{\scriptstyle\pm 0.36}$	$1.45{\scriptstyle\pm 0.38}$	$1.14{\scriptstyle\pm 0.36}$

Appendix

Table 13: Two-fidelity benchmarks: normalised mean PDF (

\uparrow

, higher is better;

>1

beats HF-only baseline). Mean

\pm

std of the per-seed paired ratio across 25 seeds. Best per row in bold . RMSE counterpart: Table 11 ; combined-mean main-body view: Table 1 .

Test Function	MAST	AR1	Recur.	Reif.	BoTorch	NARGP	MF-DGP	MISO	MFBNN
Branin (2D)	$1.24{\scriptstyle\pm 0.47}$	$0.68{\scriptstyle\pm 0.51}$	$0.63{\scriptstyle\pm 0.37}$	$1.12{\scriptstyle\pm 0.27}$	$\mathbf{1.55{\scriptstyle\pm 0.50}}$	$0.43{\scriptstyle\pm 0.28}$	$0.95{\scriptstyle\pm 0.30}$	$1.20{\scriptstyle\pm 0.30}$	$0.91{\scriptstyle\pm 0.40}$
Hartmann3 (3D)	$1.60{\scriptstyle\pm 0.48}$	$2.24{\scriptstyle\pm 0.74}$	$0.85{\scriptstyle\pm 0.26}$	$1.61{\scriptstyle\pm 0.37}$	$\mathbf{2.28{\scriptstyle\pm 0.47}}$	$1.27{\scriptstyle\pm 0.42}$	$0.63{\scriptstyle\pm 0.16}$	$1.85{\scriptstyle\pm 0.44}$	$0.80{\scriptstyle\pm 0.25}$
Ackley (4D)	$\mathbf{1.37{\scriptstyle\pm 0.27}}$	$0.77{\scriptstyle\pm 0.18}$	$0.91{\scriptstyle\pm 0.22}$	$1.05{\scriptstyle\pm 0.24}$	$0.82{\scriptstyle\pm 0.18}$	$0.78{\scriptstyle\pm 0.16}$	$0.65{\scriptstyle\pm 0.16}$	$1.19{\scriptstyle\pm 0.23}$	$0.68{\scriptstyle\pm 0.20}$
Park1 (4D)	$\mathbf{1.21{\scriptstyle\pm 0.25}}$	$1.09{\scriptstyle\pm 0.37}$	$0.59{\scriptstyle\pm 0.31}$	$0.54{\scriptstyle\pm 0.39}$	$0.61{\scriptstyle\pm 0.07}$	$0.92{\scriptstyle\pm 0.31}$	$0.23{\scriptstyle\pm 0.05}$	$0.53{\scriptstyle\pm 0.09}$	$0.43{\scriptstyle\pm 0.16}$
Park2 (4D)	$1.37{\scriptstyle\pm 0.28}$	$\mathbf{1.82{\scriptstyle\pm 0.49}}$	$1.07{\scriptstyle\pm 0.44}$	$0.21{\scriptstyle\pm 0.07}$	$1.09{\scriptstyle\pm 0.10}$	$1.17{\scriptstyle\pm 0.37}$	$0.43{\scriptstyle\pm 0.07}$	$0.13{\scriptstyle\pm 0.05}$	$0.56{\scriptstyle\pm 0.11}$
Hartmann6 (6D)	$1.61{\scriptstyle\pm 0.53}$	$1.30{\scriptstyle\pm 0.53}$	$1.11{\scriptstyle\pm 0.48}$	$\mathbf{2.30{\scriptstyle\pm 4.97}}$	$1.21{\scriptstyle\pm 0.35}$	$1.07{\scriptstyle\pm 0.54}$	$0.83{\scriptstyle\pm 0.34}$	$1.53{\scriptstyle\pm 0.43}$	$1.03{\scriptstyle\pm 0.33}$

Appendix

Table 14: Three-fidelity benchmarks: normalised mean PDF (

\uparrow

, higher is better;

>1

beats HF-only baseline). Mean

\pm

std of the per-seed paired ratio across 25 seeds. Best per row in bold . RMSE counterpart: Table 12 ; combined-mean main-body view: Table 2 .

Test function	MAST	AR1	Recur.	Reif.	BoTorch	NARGP	MF-DGP	MISO	MFBNN
Branin (2D)	$0.97{\scriptstyle\pm 1.66}$	$-0.68{\scriptstyle\pm 2.72}$	$-93.91{\scriptstyle\pm 260.58}$	$-0.87{\scriptstyle\pm 0.88}$	$\mathbf{1.41{\scriptstyle\pm 2.61}}$	$0.37{\scriptstyle\pm 1.51}$	$-0.67{\scriptstyle\pm 5.71}$	$-0.82{\scriptstyle\pm 0.86}$	$1.00{\scriptstyle\pm 1.83}$
Hartmann3 (3D)	$1.27{\scriptstyle\pm 0.24}$	$\mathbf{1.43{\scriptstyle\pm 0.25}}$	$0.81{\scriptstyle\pm 0.31}$	$1.38{\scriptstyle\pm 0.25}$	$1.42{\scriptstyle\pm 0.25}$	$1.32{\scriptstyle\pm 0.29}$	$1.08{\scriptstyle\pm 0.52}$	$1.38{\scriptstyle\pm 0.25}$	$0.98{\scriptstyle\pm 0.27}$
Ackley (4D)	$\mathbf{1.83{\scriptstyle\pm 3.93}}$	$-2.35{\scriptstyle\pm 8.56}$	$0.94{\scriptstyle\pm 1.67}$	$0.94{\scriptstyle\pm 2.25}$	$1.50{\scriptstyle\pm 2.43}$	$-1.64{\scriptstyle\pm 4.46}$	$-0.55{\scriptstyle\pm 1.16}$	$1.23{\scriptstyle\pm 2.77}$	$-0.86{\scriptstyle\pm 3.12}$
Park1 (4D)	$\mathbf{1.00{\scriptstyle\pm 0.01}}$	$1.00{\scriptstyle\pm 0.03}$	$0.66{\scriptstyle\pm 1.08}$	$0.98{\scriptstyle\pm 0.01}$	$1.00{\scriptstyle\pm 0.01}$	$1.00{\scriptstyle\pm 0.01}$	$0.96{\scriptstyle\pm 0.07}$	$0.98{\scriptstyle\pm 0.01}$	$1.00{\scriptstyle\pm 0.01}$
Park2 (4D)	$1.01{\scriptstyle\pm 0.01}$	$\mathbf{1.02{\scriptstyle\pm 0.01}}$	$0.98{\scriptstyle\pm 0.02}$	$0.86{\scriptstyle\pm 0.03}$	$1.02{\scriptstyle\pm 0.01}$	$0.98{\scriptstyle\pm 0.04}$	$0.98{\scriptstyle\pm 0.04}$	$0.87{\scriptstyle\pm 0.03}$	$1.01{\scriptstyle\pm 0.01}$
Hartmann6 (6D)	$\mathbf{3.69{\scriptstyle\pm 15.86}}$	$-2.55{\scriptstyle\pm 16.08}$	$0.94{\scriptstyle\pm 5.20}$	$-0.47{\scriptstyle\pm 12.37}$	$-2.32{\scriptstyle\pm 17.25}$	$-0.56{\scriptstyle\pm 8.02}$	$2.63{\scriptstyle\pm 15.49}$	$-0.52{\scriptstyle\pm 12.97}$	$2.42{\scriptstyle\pm 8.37}$

Appendix

Table 15: Two-fidelity benchmarks: normalised

R^{2}

(

\uparrow

, higher is better;

>1

beats HF-only baseline). Mean

\pm

std of the per-seed paired ratio across 25 seeds. Best per row in bold . Concrete and DrivAerNet are two-fidelity real-world benchmarks (HF/LF, 70/30 split, independent-sampling configuration). RMSE / Mean PDF counterparts: Tables 11 , 13 ; combined-mean main-body view: Table 1 .

Test function	MAST	AR1	Recur.	Reif.	BoTorch	NARGP	MF-DGP	MISO	MFBNN
Branin (2D)	$1.20{\scriptstyle\pm 2.22}$	$-1.05{\scriptstyle\pm 3.94}$	$-0.39{\scriptstyle\pm 2.59}$	$-0.02{\scriptstyle\pm 1.20}$	$1.64{\scriptstyle\pm 2.77}$	$-0.94{\scriptstyle\pm 2.83}$	$\mathbf{2.91{\scriptstyle\pm 9.84}}$	$0.20{\scriptstyle\pm 1.39}$	$0.96{\scriptstyle\pm 2.15}$
Hartmann3 (3D)	$1.41{\scriptstyle\pm 0.72}$	$1.55{\scriptstyle\pm 0.83}$	$1.11{\scriptstyle\pm 0.79}$	$-3.12\!\times\!10^{2}{\scriptstyle\pm 1.53\!\times\!10^{3}}$	$\mathbf{1.66{\scriptstyle\pm 0.82}}$	$1.18{\scriptstyle\pm 0.81}$	$1.25{\scriptstyle\pm 0.77}$	$1.52{\scriptstyle\pm 0.72}$	$0.57{\scriptstyle\pm 1.01}$
Ackley (4D)	$\mathbf{2.13{\scriptstyle\pm 3.59}}$	$-2.15{\scriptstyle\pm 6.49}$	$1.30{\scriptstyle\pm 3.52}$	$-2.58{\scriptstyle\pm 8.61}$	$1.65{\scriptstyle\pm 2.32}$	$0.56{\scriptstyle\pm 2.11}$	$-0.96{\scriptstyle\pm 2.49}$	$1.75{\scriptstyle\pm 3.00}$	$-2.58{\scriptstyle\pm 12.83}$
Park1 (4D)	$\mathbf{1.00{\scriptstyle\pm 0.01}}$	$1.00{\scriptstyle\pm 0.01}$	$0.98{\scriptstyle\pm 0.03}$	$0.82{\scriptstyle\pm 0.62}$	$1.00{\scriptstyle\pm 0.00}$	$0.99{\scriptstyle\pm 0.04}$	$0.98{\scriptstyle\pm 0.02}$	$0.99{\scriptstyle\pm 0.01}$	$0.97{\scriptstyle\pm 0.03}$
Park2 (4D)	$1.01{\scriptstyle\pm 0.01}$	$1.01{\scriptstyle\pm 0.01}$	$1.00{\scriptstyle\pm 0.04}$	$0.90{\scriptstyle\pm 0.02}$	$\mathbf{1.01{\scriptstyle\pm 0.01}}$	$0.99{\scriptstyle\pm 0.02}$	$0.98{\scriptstyle\pm 0.03}$	$0.90{\scriptstyle\pm 0.02}$	$0.98{\scriptstyle\pm 0.02}$
Hartmann6 (6D)	$0.71{\scriptstyle\pm 8.09}$	$4.64{\scriptstyle\pm 24.15}$	$1.22{\scriptstyle\pm 5.82}$	$0.36{\scriptstyle\pm 129.44}$	$0.40{\scriptstyle\pm 17.09}$	$1.91{\scriptstyle\pm 13.19}$	$\mathbf{5.70{\scriptstyle\pm 23.05}}$	$0.14{\scriptstyle\pm 10.10}$	$1.58{\scriptstyle\pm 15.21}$

Appendix

Table 16: Three-fidelity benchmarks: normalised

R^{2}

(

\uparrow

, higher is better;

>1

beats HF-only baseline). Mean

\pm

std of the per-seed paired ratio across 25 seeds. Best per row in bold . HOIP is the three-fidelity real-world benchmark (HF/MF/LF, 50/30/20 split). Large magnitudes (e.g. Rastrigin row, HOIP row) reflect a near-zero HF-only denominator, not exceptionally good fits. RMSE / Mean PDF counterparts: Tables 12 , 14 ; combined-mean main-body view: Table 2 .

	Normalised RMSE $\downarrow$									Normalised Mean PDF $\uparrow$
Benchmark	MAST	AR1	Recur.	Reif.	BoTorch	NARGP	MF-DGP	MISO	MFBNN	MAST	AR1	Recur.	Reif.	BoTorch	NARGP	MF-DGP	MISO	MFBNN
HOIP (3D)	$1.08{\scriptstyle\pm 0.16}$	$2.37{\scriptstyle\pm 2.67}$	$\mathbf{1.00{\scriptstyle\pm 0.25}}$	$2.12{\scriptstyle\pm 0.46}$	$1.39{\scriptstyle\pm 0.36}$	$1.12{\scriptstyle\pm 0.24}$	$1.20{\scriptstyle\pm 0.34}$	$2.12{\scriptstyle\pm 0.46}$	$1.74{\scriptstyle\pm 0.49}$	$\mathbf{1.12{\scriptstyle\pm 0.28}}$	$0.86{\scriptstyle\pm 1.79}$	$1.03{\scriptstyle\pm 0.27}$	$0.63{\scriptstyle\pm 0.17}$	$0.68{\scriptstyle\pm 0.21}$	$0.90{\scriptstyle\pm 0.24}$	$0.81{\scriptstyle\pm 0.22}$	$0.63{\scriptstyle\pm 0.17}$	$0.66{\scriptstyle\pm 0.24}$
DrivAerNet (23D)	$1.01{\scriptstyle\pm 0.10}$	$1.27{\scriptstyle\pm 0.16}$	$2.73{\scriptstyle\pm 3.20}$	$\mathbf{0.98{\scriptstyle\pm 0.01}}$	$1.24{\scriptstyle\pm 0.07}$	$1.04{\scriptstyle\pm 0.12}$	$0.99{\scriptstyle\pm 0.08}$	$0.98{\scriptstyle\pm 0.01}$	$7.34{\scriptstyle\pm 1.12}$	$\mathbf{1.02{\scriptstyle\pm 0.09}}$	$0.97{\scriptstyle\pm 0.11}$	$0.15{\scriptstyle\pm 0.29}$	$0.60{\scriptstyle\pm 0.03}$	$0.45{\scriptstyle\pm 0.05}$	$0.96{\scriptstyle\pm 0.08}$	$0.73{\scriptstyle\pm 0.17}$	$0.60{\scriptstyle\pm 0.03}$	$0.09{\scriptstyle\pm 0.02}$

Appendix

Table 17: HOIP and DrivAerNet under collocated sampling: normalised RMSE / normalised mean PDF, mean

\pm

std across 25 seeds. All values are normalised relative to the cost-equivalent HF-only baseline. Best per row and metric in bold .

	Normalised RMSE $\downarrow$		Normalised mean PDF $\uparrow$
Test function	MAST (GP)	MAST-ICL	MAST (GP)	MAST-ICL
Branin (2D)	$1.19{\scriptstyle\pm 0.40}$	$\mathbf{1.04{\scriptstyle\pm 0.31}}$	$\mathbf{1.28{\scriptstyle\pm 0.43}}$	$0.24{\scriptstyle\pm 0.08}$
Ackley (4D)	$\mathbf{0.94{\scriptstyle\pm 0.26}}$	$0.95{\scriptstyle\pm 0.28}$	$\mathbf{1.23{\scriptstyle\pm 0.32}}$	$0.27{\scriptstyle\pm 0.04}$
Park1 (4D)	$\mathbf{0.97{\scriptstyle\pm 0.48}}$	$1.07{\scriptstyle\pm 0.38}$	$\mathbf{1.46{\scriptstyle\pm 0.49}}$	$0.21{\scriptstyle\pm 0.03}$
Hartmann6 (6D)	$0.88{\scriptstyle\pm 0.19}$	$\mathbf{0.77{\scriptstyle\pm 0.12}}$	$\mathbf{1.80{\scriptstyle\pm 0.52}}$	$0.51{\scriptstyle\pm 0.18}$
Rosenbrock (20D)	$\mathbf{0.37{\scriptstyle\pm 0.04}}$	$0.45{\scriptstyle\pm 0.04}$	$\mathbf{3.23{\scriptstyle\pm 0.23}}$	$0.40{\scriptstyle\pm 0.03}$

Appendix

Table 18: Preliminary MAST-ICL results on the two-fidelity benchmarks where MAST-ICL is competitive with or improves on MAST-GP. Normalised RMSE (lower is better) and normalised mean PDF (higher is better), mean

\pm

std across 25 seeds. Best of the two columns per metric in bold .

Surrogate Modeling

Surrogate modeling builds faster or more interpretable approximations of expensive simulations and black-box models.

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Mar 23, 2026

Multifidelity-Augmented Gaussian Process Inputs for Surrogate Modeling from Scarce Data

Supervised machine learning describes the practice of fitting a parameterized model to labeled input-output data. Supervised machine learning methods have demonstrated promise in learning efficient surrogate models that can (partially) replace expensive high-fidelity models, making many-query analyses, such as optimization, uncertainty quantification, and inference, tractable. However, when training data must be obtained through the evaluation of an expensive model or experiment, the amount of training data that can be obtained is often limited, which can make learned surrogate models unreliable. In many engineering and scientific settings, cheaper low-fidelity models may be available, for example arising from simplified physics modeling or coarse grids. These models may be used to generate additional low-fidelity training data. The goal of multifidelity machine learning is to use both high- and low-fidelity training data to learn a surrogate model which is cheaper to evaluate than the high-fidelity model, but more accurate than any available low-fidelity model. This work proposes a new multifidelity training approach for Gaussian process regression which uses low-fidelity data to define additional features that augment the input space of the learned model. Similarly to cokriging estimators, the proposed approach conditions the high-fidelity surrogate model on the predictions of all available low-fidelity surrogate models, while benefiting from the computational efficiency of autoregressive estimators. Numerical experiments on several test problems demonstrate both increased predictive accuracy and reduced computational cost relative to the state of the art.

Atticus Rex, Elizabeth Qian, David Peterson

School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA · School of Computational Science & Engineering, Georgia Institute of Technology, Atlanta, GA · Air Force Research Lab, Wright-Patterson AFB, Dayton, Ohio

Surrogate Modeling Multifidelity Modeling Gaussian Process Regression

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Mar 10, 2026

A Unified Hierarchical Multi-Task Multi-Fidelity Framework for Data-Efficient Surrogate Modeling in Manufacturing

Surrogate modeling is an essential data-driven technique for quantifying relationships between input variables and system responses in manufacturing and engineering systems. Two major challenges limit its effectiveness: (1) large data requirements for learning complex nonlinear relationships, and (2) heterogeneous data collected from sources with varying fidelity levels. Multi-task learning (MTL) addresses the first challenge by enabling information sharing across related processes, while multi-fidelity modeling addresses the second by accounting for fidelity-dependent uncertainty. However, existing approaches typically address these challenges separately, and no unified framework simultaneously leverages inter-task similarity and fidelity-dependent data characteristics. This paper develops a novel hierarchical multi-task multi-fidelity (H-MT-MF) framework for Gaussian process-based surrogate modeling. The proposed framework decomposes each task's response into a task-specific global trend and a residual local variability component that is jointly learned across tasks using a hierarchical Bayesian formulation. The framework accommodates an arbitrary number of tasks, design points, and fidelity levels while providing predictive uncertainty quantification. We demonstrate the effectiveness of the proposed method using a 1D synthetic example and a real-world engine surface shape prediction case study. Compared to (1) a state-of-the-art MTL model that does not account for fidelity information and (2) a stochastic kriging model that learns tasks independently, the proposed approach improves prediction accuracy by up to 19% and 23%, respectively. The H-MT-MF framework provides a general and extensible solution for surrogate modeling in manufacturing systems characterized by heterogeneous data sources.

Manan Mehta, Zhiqiao Dong, Yuhang Yang +1

Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, United States · Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 41809, United States

Surrogate Modeling Multifidelity Modeling Multi-Task Learning Gaussian Process Regression

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Oct 12, 2026

Compute-optimal data scaling for neural surrogates via multi-fidelity training

Neural surrogates for Partial Differential Equations (PDEs) are usually trained on synthetic data from numerical simulators. As models grow and scaling laws demand larger datasets, the cost of data generation often outweighs the training cost and becomes the bottleneck in the scaling quest. Unlike data in other domains, however, each PDE sample has a tunable cost that depends on its numerical fidelity, with the resulting cost-accuracy trade-off governed by known convergence rates. We exploit this property by formalizing neural surrogate training as a function of both data quantity and fidelity under a fixed data generation compute budget, treating fidelity as a fine-grained discrete spectrum rather than a binary distinction. Building on this, we show that multi-fidelity training enables Pareto-optimal error scaling against data generation compute. By using simulations at lower fidelities as an additional training signal, training on the optimal fidelity mixture matches the accuracy of high-fidelity-only training, with up to

6.3\times

less data generation compute across our settings. We validate these findings across six PDE problems spanning structured grids and unstructured meshes, two distinct error sources (discretization and iterative-solver error), and four surrogate architectures, totaling over 7,000 training runs.

Paul Setinek, Fabian Paischer, Nils Thuerey +2

Institute for Machine Learning, JKU Linz · Mistral AI · Technical University of Munich +1

Neural Surrogate Modeling PDE Surrogate Modeling Multifidelity Modeling

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Abstract

Surrogate Modeling

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Multifidelity-Augmented Gaussian Process Inputs for Surrogate Modeling from Scarce Data

A Unified Hierarchical Multi-Task Multi-Fidelity Framework for Data-Efficient Surrogate Modeling in Manufacturing

Compute-optimal data scaling for neural surrogates via multi-fidelity training