# Dimensionally consistent surrogate modelling through dimensional analysis and harmonic expansions

arXiv: https://arxiv.org/abs/2609.38094 · HTML: https://mlsanity.com/paper/2609.38094/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 25 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 4 of 4)

- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Dimensionally Consistent Learning with Buckingham Pi](https://mlsanity.com/paper/2202.04643/) (2022)
- [Dimensionless machine learning: Imposing exact units equivariance](https://mlsanity.com/paper/2204.00887/) (2022)

## Abstract (arXiv)

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 Pi-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.