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Complexity reduction of physical models: an equation-free approach by means of scaling

Simone Rusconi, Christina Schenk, Razvan Ceuca, Arghir Zarnescu, Elena Akhmatskaya

TL;DR

This work proposes Generalized Optimal Scaling (GOS), a unified, equation-free framework to reduce the complexity of multi-parameter physical models by identifying a minimal set of independent dimensionless parameters and expressing model coefficients as monomials of these parameters. Building on Optimal Scaling (OS) and Optimal Scaling with Constraints (OSC), GOS provides a principled way to minimize the variation of dimensionless coefficients and to determine when terms can be dropped without significantly altering solutions, thereby mitigating over-parameterization and improving numerical conditioning. The approach is demonstrated on the classical projectile model, where GOS yields substantial reductions in coefficient variability and clear, quantitative thresholds for asymptotic simplifications across multiple regimes. The results offer a scalable path toward automated, minimal-parameter scaling for complex multiscale systems with robust calibration properties and improved numerical stability.

Abstract

The description of complex physical phenomena often involves sophisticated models that rely on a large number of parameters, with many dimensions and scales. One practical way to simplify that kind of models is to discard some of the parameters, or terms of underlying equations, thus giving rise to reduced models. Here, we propose a general approach to obtaining such reduced models. The method is independent of the model in use, i.e., equation-free, depends only on the interplay between the scales and dimensions involved in the description of the phenomena, and controls over-parametrization. It also quantifies conditions for asymptotic models by providing explicitly computable thresholds on values of parameters that allow for reducing complexity of a model, while preserving essential predictive properties. Although our focus is on complexity reduction, this approach may also help with calibration by mitigating the risks of over-parameterization and instability in parameter estimation. The benefits of this approach are discussed in the context of the classical projectile model.

Complexity reduction of physical models: an equation-free approach by means of scaling

TL;DR

This work proposes Generalized Optimal Scaling (GOS), a unified, equation-free framework to reduce the complexity of multi-parameter physical models by identifying a minimal set of independent dimensionless parameters and expressing model coefficients as monomials of these parameters. Building on Optimal Scaling (OS) and Optimal Scaling with Constraints (OSC), GOS provides a principled way to minimize the variation of dimensionless coefficients and to determine when terms can be dropped without significantly altering solutions, thereby mitigating over-parameterization and improving numerical conditioning. The approach is demonstrated on the classical projectile model, where GOS yields substantial reductions in coefficient variability and clear, quantitative thresholds for asymptotic simplifications across multiple regimes. The results offer a scalable path toward automated, minimal-parameter scaling for complex multiscale systems with robust calibration properties and improved numerical stability.

Abstract

The description of complex physical phenomena often involves sophisticated models that rely on a large number of parameters, with many dimensions and scales. One practical way to simplify that kind of models is to discard some of the parameters, or terms of underlying equations, thus giving rise to reduced models. Here, we propose a general approach to obtaining such reduced models. The method is independent of the model in use, i.e., equation-free, depends only on the interplay between the scales and dimensions involved in the description of the phenomena, and controls over-parametrization. It also quantifies conditions for asymptotic models by providing explicitly computable thresholds on values of parameters that allow for reducing complexity of a model, while preserving essential predictive properties. Although our focus is on complexity reduction, this approach may also help with calibration by mitigating the risks of over-parameterization and instability in parameter estimation. The benefits of this approach are discussed in the context of the classical projectile model.

Paper Structure

This paper contains 26 sections, 14 theorems, 108 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Employed Notation
  4. The classical projectile model: scaling strategies
  5. Optimal Scaling: Overview
  6. Assumptions and Main Results
  7. Nondimensional Setting
  8. Optimization
  9. Optimal Scaling
  10. Formulation
  11. Optimal Scaling with Constraints
  12. Formulation
  13. Generalized Optimal Scaling
  14. Formulation
  15. A Case Study: the classical projectile model
  16. ...and 11 more sections

Key Result

Theorem 1

Given prop:phys_param_dimenprop:phys_param_dimen for physical parameters $\tilde{p}_1,\dots,\tilde{p}_{N_{\mathsf{p}}}$, any quantity of the form is dimensionless if and only if the vector $\vec{z} \coloneqq ( z_1, z_2, \dots, z_{N_{\mathsf{p}}} )^{T} \in \mathbb{R}^{N_{\mathsf{p}}}$ of exponents in eqn:ZZdimless belongs to the null space (kernel) of the matrix where the entries of $\textbf{M}$e

Figures (4)

  • Figure 1: Coefficients \ref{['eqn:OS_ProgPb_lam_PI']}-\ref{['eqn:OSC_ProgPb_s4_lam_PI']} provided by $\hbox{GOS}$ as functions of $\pi_1,\pi_2$\ref{['eqn:PI_ProjPb']}. The regime \ref{['eqn:regime_unity_GOS']}-\ref{['eqn:threshold_GOS']} is achieved in the areas delimited by magenta dashed lines.
  • Figure 2: Values attained by coefficients $\lambda_1,\lambda_2,\lambda_3,\lambda_4$\ref{['eqn:lambdas_def_projectile']} and \ref{['eqn:OS_ProgPb_lam_PI']} (\ref{['fig:lambdas_val_PP_(a)']}) and corresponding variability $\mathop{\mathrm{max}}_i \lambda_i / \mathop{\mathrm{min}}_i \lambda_i$ (\ref{['fig:lambdas_val_PP_(b)']}), when $\tilde{g} = 9.81 \, \mathrm{m} \, \mathrm{s}^{-2}$, $\tilde{R} = 6.3781 \times 10^{6} \, \mathrm{m}$, $\tilde{h}_0 = 1 \, \mathrm{m}$, $\tilde{v}_0 = 1 \, \mathrm{m} \, \mathrm{s}^{-1}$ and $\pi_1,\pi_2$\ref{['eqn:PI_ProjPb']}. In both \ref{['fig:lambdas_val_PP_(a)']} and \ref{['fig:lambdas_val_PP_(b)']}, the black circles provide the numerical values obtained by Generalized Optimal Scaling ($\hbox{GOS}$) with $\mathtt{s}=0$, i.e., the coefficients $\lambda_1,\lambda_2,\lambda_3,\lambda_4$\ref{['eqn:OS_ProgPb_lam_PI']}. \ref{['fig:lambdas_val_PP_(a)']} and \ref{['fig:lambdas_val_PP_(b)']} also show the values achieved by $\lambda_1,\lambda_2,\lambda_3,\lambda_4$\ref{['eqn:lambdas_def_projectile']} if $N_{\mathsf{x}}=2$ out of the $N_{\mathsf{l}}=4$ are imposed to be equal to $1$, as indicated by the legend in \ref{['fig:lambdas_val_PP_(a)']} and \ref{['fig:lambdas_val_PP_(b)']}. The Figures do not show the data corresponding to $\lambda_2=\lambda_3=1$, since it leads to $\tilde{p}_2^{-1} \, \tilde{\theta}_2 = \tilde{p}_3 \, \tilde{\theta}_2^{-1} = 1$ and there are no $\tilde{\theta}_1$ and $\tilde{\theta}_2$ solving such a system of equations.
  • Figure 3: \ref{['fig:ref_sol_Round_PP']} shows the numerical solution $\varphi=\varphi_{\mathrm{ref}}$ to the Ordinary Differential Equation \ref{['eqn:dimensioless_ODE_projectile']} in the time interval $[0,T]$, with $\lambda_1 = g$, $\lambda_2 = R^{-1}$, $\lambda_3=h_0$, $\lambda_4=v_0$ and $T=700$, being $\tilde{g} = 9.81 \, \mathrm{m} \, \mathrm{s}^{-2}$, $\tilde{R} = 6.3781 \times 10^{6} \, \mathrm{m}$, $\tilde{h}_0 = 1 \, \mathrm{m}$, $\tilde{v}_0 = 1 \, \mathrm{m} \, \mathrm{s}^{-1}$. Given the shown $\varphi_{\mathrm{ref}}$ and $\tilde{g}$, $\tilde{R}$, $\tilde{h}_0$, $\tilde{v}_0$ as above, \ref{['fig:en_Round_PP']} depicts the error $e_{n}$\ref{['eqn:error_Round_PP']} corresponding to several choices of the coefficients $\lambda_1,\lambda_2,\lambda_3,\lambda_4$, as indicated in the legend. The black circles show the error made by using $\lambda_1,\lambda_2,\lambda_3,\lambda_4$\ref{['eqn:OS_ProgPb_lam_PI']} with $\pi_1,\pi_2$\ref{['eqn:PI_ProjPb']}, as provided by Generalized Optimal Scaling ($\hbox{GOS}$) with $\mathtt{s}=0$. On the other hand, the colored symbols indicate the error corresponding to numerical values of $\lambda_1,\lambda_2,\lambda_3,\lambda_4$ obtained by imposing $N_{\mathsf{x}}=2$ out of the $N_{\mathsf{l}}=4$ coefficients $\lambda_1,\lambda_2,\lambda_3,\lambda_4$\ref{['eqn:lambdas_def_projectile']} to be equal to $1$. \ref{['fig:en_Round_PP']} does not show the data corresponding to $\lambda_2=\lambda_3=1$, since it leads to $\tilde{p}_2^{-1} \, \tilde{\theta}_2 = \tilde{p}_3 \, \tilde{\theta}_2^{-1} = 1$ and there are no $\tilde{\theta}_1$ and $\tilde{\theta}_2$ solving such a system of equations.
  • Figure 4: Error $\epsilon_{\mathtt{s}}(T)$\ref{['eqn:OSC_ProgPb_approx_error']} and coefficient $\lambda_\mathtt{s}$\ref{['eqn:OSC_ProgPb_lams_small']} as functions of $\pi_1,\pi_2$\ref{['eqn:PI_ProjPb']}. Along the white solid and dotted lines, the error $\epsilon_{\mathtt{s}}(T)$ is equal to $10^{-1}$ and $10^{-2}$, respectively. The inequalities \ref{['eqn:OSC_ProgPb_lams_small']} are met with $\delta=-2$ and $\gamma=1/2$ in the areas delimited by magenta dashed lines.

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 7
  • proof
  • Theorem 7
  • ...and 11 more