Dimensionally consistent surrogate modelling through dimensional analysis and harmonic expansions

Ernest Tarrus; Hector Gisbert

cs.LGSep 29, 2026

Dimensionally consistent surrogate modelling through dimensional analysis and harmonic expansions

Authors: Ernest Tarrus, Hector Gisbert

Organizations: Universidad Europea de Valencia, Escuela de Ciencias, Ingenier´ıa y Dise˜no, C/ de Guillem de Castro, 175, Extramurs, Valencia, 46008, Valencia, Spain

Abstract

Dimensional homogeneity is a fundamental constraint on physically meaningful models, requiring invariance under changes of units. We present a data-driven method for constructing surrogate models that satisfy this constraint at the level of the hypothesis class. Starting from a dimension matrix of measured variables, the method derives Buckingham $Π$ -groups, constructs admissible dimensional prefactors, and approximates the remaining dimensionless dependence using truncated harmonic expansions on normalized invariant domains. Once the prefactor and dictionary are fixed, the coefficients are obtained from a regularized linear regression problem. We test the approach on the simple pendulum, Planck's black-body law, the double-pendulum Lyapunov field, and an experimental COBE/FIRAS black-body spectrum dataset. The results show that dimensional constraints improve conditioning, robustness to noise, and sample efficiency relative to unconstrained baselines, while the choice of dictionary becomes important in non-periodic or multi-invariant settings. The learned expressions are explicit and inexpensive to evaluate, which makes them useful as surrogate models for structured physical problems.

Figures & tables

Panel of Figure 1 — Figure 1: Algorithmic pipeline. Box 2c illustrates the two harmonic dictionary formulations evaluated in the experiments: separable tensor-product (option $A$ ) and phase-combination (option $B$ ) dictionaries.

\hat{\mathbf{w}}_{\bm{\beta}}=\operatorname*{argmin}_{\mathbf{w}}\left\|\mathbf{y}-\mathcal{X}_{\bm{\beta}}\mathbf{w}\right\|_{2}^{2}+\mu\|\mathbf{w}\|_{2}^{2}

Algorithm 1 Unit-equivariant learning via tensor-product harmonic dictionaries

Scenario	$\sigma_{\mathrm{abs}}$	$(n_{L},M)$	Method	Coeff. $(c)$	Intercept $(d)$	Error (%)
Low Noise	0.02	(5,5)	RidgeCV	6.2539	0.0061	0.47
Low Noise	0.02	(5,5)	Minuit	$6.2451\pm 0.0287$	$0.0079\pm 0.0011$	0.61
High Density	0.02	(50,50)	RidgeCV	6.2836	0.0007	0.01
High Density	0.02	(50,50)	Minuit	$6.2836\pm 0.0142$	$0.0006\pm 0.0053$	0.01
High Noise	0.6	(5,5)	RidgeCV	5.4052	0.1822	13.97
High Noise	0.6	(5,5)	Minuit	$5.1418\pm 0.8622$	$0.2382\pm 0.3164$	18.16

Table 1: Simple-pendulum results comparing regularized regression and Minuit . The theoretical small-angle coefficient is

2\pi\approx 6.2832

Panel of Figure 2 — Figure 2: Regression results for the simple pendulum period under different noise and sampling regimes. The red dashed line indicates the theoretical small-angle coefficient $2\pi$ .

$n_{L}$	$R^{2}$ (Dim.)	$R^{2}$ (Unc.)	$\hat{\alpha}\pm\sigma_{\alpha}$
5	0.9982	—	—
8	0.9977	—	—
12	0.9985	0.3061	$0.796\pm 0.140$
20	0.9991	0.9667	$0.678\pm 0.132$
35	0.9995	0.9930	$0.556\pm 0.099$
60	0.9994	0.9983	$0.528\pm 0.034$

Table 2: Dimensional versus unconstrained power-law regression for the simple pendulum. Mean out-of-sample

R^{2}

and recovered exponent

\hat{\alpha}

over 30 independent trials. Both models fitted with Minuit .

Figure 3: Dimensional vs. unconstrained power-law baseline on the simple pendulum. Left: out-of-sample $R^{2}$ as a function of $n_{L}$ . Right: recovered exponent $\hat{\alpha}$ vs. $n_{L}$ , skipping $n_{L}<20$ ; the dashed line marks the dimensional value $\alpha=1/2$ . Error bars denote $\pm 1$ standard deviation over 30 independent trials. Both models fitted with Minuit .

$N_{\mathrm{dat}}$	Best $\hat{\alpha}$	Best $R^{2}$	Best MSE
50	1.0	0.999938	9.707071e-20
600	1.0	0.999970	4.605817e-20
1150	0.0	0.999998	3.589867e-21
1700	0.0	0.999998	3.440877e-21
2250	0.0	0.999998	3.339133e-21
2800	0.0	0.999998	3.255943e-21

Table 3: Black-body Regime A: broad prefactor grid (

\alpha\in[-5,5]

), low truncation (

N_{f}=5

$N_{\mathrm{dat}}$	Best $\hat{\alpha}$	Best $R^{2}$	Best MSE
50	0.5	1.0	6.046916e-28
600	0.0	1.0	2.807961e-28
1150	0.0	1.0	2.365705e-28
1700	0.0	1.0	2.157930e-28
2250	0.0	1.0	2.016073e-28
2800	0.0	1.0	1.906302e-28

Table 4: Black-body Regime B: broad prefactor grid (

\alpha\in[-5,5]

), high truncation (

N_{f}=60

$N_{\mathrm{dat}}$	Best $\hat{\alpha}$	Best $R^{2}$	Best MSE
50	0.05	0.999879	1.894525e-19
600	-0.05	0.999977	3.485987e-20
1150	0.00	0.999998	3.589867e-21
1700	0.00	0.999998	3.440877e-21
2250	0.00	0.999998	3.339133e-21
2800	0.00	0.999998	3.255943e-21

Table 5: Black-body Regime C: narrow prefactor grid (

\alpha\in[-0.5,0.5]

), low truncation (

N_{f}=5

$N_{\mathrm{dat}}$	Best $\hat{\alpha}$	Best $R^{2}$	Best MSE
50	0.50	1.0	6.046916e-28
600	0.00	1.0	2.807961e-28
1150	0.00	1.0	2.365705e-28
1700	-0.10	1.0	2.033666e-28
2250	-0.05	1.0	1.958330e-28
2800	0.00	1.0	1.906302e-28

Table 6: Black-body Regime D: narrow prefactor grid (

\alpha\in[-0.5,0.5]

), high truncation (

N_{f}=60

Figure 4: Regime B ( $N_{f}=60$ , $\alpha\in[-5,5]$ ), low noise ( $\sigma_{\mathrm{rel}}=0.02$ , $N_{\text{dat}}=5000$ ): near-perfect reconstruction of the Planck curve.

Figure 5: Example of functional overfitting in the high-noise regime (Regime B, $\sigma_{\mathrm{rel}}=0.6$ , $N_{f}=60$ ). The prefactor scan still identifies the correct dimensional scaling, but the fitted modulation begins to track noise fluctuations.

Figure 6: Fit of the dimensionally constrained model to COBE/FIRAS CMB monopole spectrum data (43 points, $\alpha=0$ , $N_{f}=30$ ). Left: spectral radiance $B_{\nu}$ as a function of frequency; the theoretical Planck curve at $T=2.725$ K (dashed) is overlaid with the FIRAS measurements (blue circles) and the dimensional fit (red line). Right: pull distribution $(B_{\nu}-\hat{B}_{\nu})/\sigma$ in units of the FIRAS measurement uncertainty $\sigma$ .

Figure 7: Bootstrap distribution of the recovered prefactor exponent $\hat{\alpha}$ over 500 classic bootstrap samples (with replacement) of the 43 FIRAS data points, using $N_{f}=8$ harmonic terms. The theoretical value $\alpha=0$ (solid black line) and the bootstrap result $\hat{\alpha}=0.15\pm 0.36$ (dashed red line) are indicated. The distribution is concentrated near zero with a modest spread ( $\sigma\approx 0.36$ ).

Figure 8: Comparison of harmonic and Chebyshev polynomial bases on the black-body benchmark. Left: out-of-sample $R^{2}$ as a function of the truncation order $N_{f}$ , with both bases using the same number of features ( $1+2N_{f}$ ). Right: sample efficiency at fixed $N_{f}=8$ , showing that the harmonic expansion generalizes reliably from as few as 20 training points while Chebyshev requires at least 100.

$\sigma_{\text{rel}}$	Harmonic (ours)	MLP on $\pi_{1}$ (Villar-style baseline)
0.00	1.000000	0.987803
0.02	0.999652	0.987709
0.10	0.990905	0.980457
0.30	0.923498	0.891541
0.60	0.721088	0.683265

Table 7: Comparison between the proposed harmonic model and a Villar et al.–style MLP baseline on the black-body benchmark (

N_{f}=8

, 70/30 train/test split).

Figure 9: Comparison of our harmonic expansion method against a Villar et al.style MLP operating on the same dimensionless invariant space. Left: out-of-sample $R^{2}$ as a function of training dataset size (noiseless). Right: out-of-sample $R^{2}$ under increasing additive noise ( $N=200$ ). The harmonic model matches or exceeds the MLP in these tests while using a linear fit in the coefficients.

Truncation ( $N_{f}$ )	$R^{2}$	MSE
5	0.791778	0.517102
20	0.816322	0.456150
30	0.823415	0.438534

Table 8: Performance of the phase-combination dictionary for the symmetric baseline configuration (

m_{1}=m_{2}

l_{1}=l_{2}

Truncation ( $N_{f}$ )	$R^{2}$	MSE
5	0.860839	0.345595
20	0.933313	0.165611

Table 9: Performance of the separable tensor-product dictionary for the symmetric baseline configuration (

m_{1}=m_{2}

l_{1}=l_{2}

Term	Value
$C_{1,0}$ (cos-cos)	-7.976002
$S_{1,1}$ (sin-sin)	-3.137987
$C_{0,1}$ (cos-cos)	-2.709306
$S_{1,2}$ (sin-sin)	2.003437
$S_{3,2}$ (sin-sin)	1.498303
$C_{2,3}$ (cos-cos)	-1.259975

Table 10: Ten largest coefficients for the separable model at

N_{f}=20

Dictionary Type	$N_{f}$	$R^{2}$	MSE
Separable + Parity	20	0.933290	0.165668
Phase-Comb + Parity	40	0.826657	0.430482
Phase-Comb + Parity	50	0.828744	0.425301
Separable + Parity	35	0.950272	0.123495

Table 11: Performance of parity-reduced dictionaries for the symmetric baseline configuration (

m_{1}=m_{2}

l_{1}=l_{2}

Term	Value
$C_{1,0}$ (cos-cos)	-7.972290
$S_{1,1}$ (sin-sin)	-3.137987
$C_{0,1}$ (cos-cos)	-2.708697
$S_{1,2}$ (sin-sin)	2.003437
$S_{3,2}$ (sin-sin)	1.498303
$C_{2,3}$ (cos-cos)	-1.260299

Table 12: Ten largest coefficients for the parity-reduced separable model at

N_{f}=35

Panel of Figure 10 — Figure 10: Progressive improvement in the reconstruction of the double-pendulum Lyapunov field. The separable tensor-product dictionary is essential for resolving the anisotropic angular structure, while parity reduction improves efficiency without degrading accuracy.

Dictionary Type	$N_{f}$	$R^{2}$	MSE
Phase-Combination	5	0.704248	3.395539
Phase-Combination	20	0.717002	3.249113
Separable + Parity	5	0.908818	1.046867
Separable + Parity	20	0.942478	0.660411

Table 13: Results for the asymmetric-mass double-pendulum configuration.

Panel of Figure 11 — Figure 11: Reconstruction quality for the asymmetric double-pendulum configuration. The separable tensor-product basis again provides a substantially better approximation than the phase-combination basis.

Figure 12: Comparison of separable harmonic and separable Chebyshev dictionaries on the double-pendulum Lyapunov field. Left: ground truth. Center: harmonic fit at $N_{f}=8$ . Right: Chebyshev fit at $N_{f}=8$ . The harmonic basis captures both marginal and interaction angular structure more faithfully.

Figure 13: Train and test $R^{2}$ as a function of the truncation order $N_{f}$ for the parity-reduced separable model on an 80/20 split of 8000 subsampled grid points. Test performance peaks at $N_{f}\approx 30$ and decreases at higher $N_{f}$ due to overfitting, indicating that the plateau reported in the earlier tables is a consequence of finite training samples rather than a limitation of the harmonic representation.

Figure 14: Heatmaps of the absolute prediction error $|\hat{\lambda}_{\max}-\lambda_{\max}|$ for the parity-reduced separable model at three truncation orders. Errors concentrate at the boundaries between regular and chaotic regions and narrow progressively with increasing $N_{f}$ , but the interface structure persists.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 15: Summary of the noisy black-body experiments across the four regimes. Top: recovered prefactor exponent $\hat{\alpha}$ vs. $N_{\mathrm{dat}}$ . Bottom: best $R^{2}$ vs. $N_{\mathrm{dat}}$ . Three noise levels are shown per panel. The prefactor exponent remains robust across all regimes, while $R^{2}$ degrades with noise and, for large- $N_{f}$ models, also with small $N_{\mathrm{dat}}$ .

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Surrogate models provide compact relations between user-defined input parameters and output quantities of interest, enabling the efficient evaluation of complex parametric systems in many-query settings. Such capabilities are essential in a wide range of applications, including optimisation, control, data assimilation, uncertainty quantification, and emerging digital twin technologies in various fields such as manufacturing, personalised healthcare, smart cities, and sustainability. This article reviews established methodologies for constructing surrogate models exploiting either knowledge of the governing laws and the dynamical structure of the system (physics-based) or experimental observations (data-driven), as well as hybrid approaches combining these two paradigms. By revisiting the design of a surrogate model as a functional approximation problem, existing methodologies are reviewed in terms of the choice of (i) a reduced basis and (ii) a suitable approximation criterion. The paper reviews methodologies pertaining to the field of Scientific Machine Learning, and it aims at synthesising established knowledge, recent advances, and new perspectives on: dimensionality reduction, physics-based, and data-driven surrogate modelling based on proper orthogonal decomposition, proper generalised decomposition, and artificial neural networks; multi-fidelity methods to exploit information from sources with different fidelities; adaptive sampling, enrichment, and data augmentation techniques to enhance the quality of surrogate models.

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