Dimensionally consistent models improve physical data predictions
Dimensionally consistent surrogate modelling through dimensional analysis and harmonic expansions
Machine Learning
Summary
Physical models need to work correctly no matter what units you use, like meters or feet. The authors created a method that builds these models by combining known physics rules about units with math tools to fit data more efficiently. Their approach makes models that are simple, reliable, and better at handling noisy or limited data. They tested it on examples like pendulums and black-body radiation to show it works well.
What this means in practice
- •For engineering simulation teams: Create simplified yet accurate surrogate models for complex physical systems that respect unit invariance to improve simulation robustness and efficiency.
- •For environmental monitoring teams: Build robust models for interpreting physical measurement data affected by noise using dimensionally consistent surrogates, enhancing reliability of environmental predictions.
Authors
Ernest Tarrus, Hector Gisbert
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.