# Physics-Informed Regression: Parameter Estimation in Parameter-Linear Nonlinear Dynamic Models

arXiv: https://arxiv.org/abs/2508.19249 · HTML: https://mlsanity.com/paper/2508.19249/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 56 references that are in the map.
- First posted on arXiv 25 Jul 2025.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [SoftAdapt: Techniques for Adaptive Loss Weighting of Neural Networks with Multi-Part Loss Functions](https://mlsanity.com/paper/1912.12355/) (2019)
- [Data-driven approaches for predicting spread of infectious diseases through DINNs: Disease Informed Neural Networks](https://mlsanity.com/paper/2110.05445/) (2021)
- [Physics-informed neural networks for one-dimensional sound field predictions with parameterized sources and impedance boundaries](https://mlsanity.com/paper/2109.11313/) (2021)
- [Physics-informed neural networks for parameter learning of wildfire spreading](https://mlsanity.com/paper/2406.14591/) (2024)
- [Collocation methods for second and higher order systems](https://mlsanity.com/paper/2302.09056/) (2023)

## Built on by (most cited first, 2 of 2)

- [DASyR-LLM: Domain-Aware Symbolic Regression with LLMs for Kinetic Model Discovery](https://mlsanity.com/paper/2608.05120/) (2026)
- [Conformalized Physics-Informed Weak Regression for Parameter Estimation in Epidemic Models](https://mlsanity.com/paper/2610.19058/) (2026)

## Abstract (arXiv)

We present a new efficient hybrid parameter estimation method based on the idea, that if nonlinear dynamic models are stated in terms of a system of equations that is linear in terms of the parameters, then regularized ordinary least squares can be used to estimate these parameters from time series data. We introduce the term "Physics-Informed Regression" (PIR) to describe the proposed data-driven hybrid technique as a way to bridge theory and data by use of ordinary least squares to efficiently perform parameter estimation of the model coefficients of different parameter-linear models; providing examples of models based on nonlinear ordinary equations (ODE) and partial differential equations (PDE). The focus is on parameter estimation on a selection of ODE and PDE models, each illustrating performance in different model characteristics. For two relevant epidemic models of different complexity and number of parameters, PIR is tested and compared against the related technique, physics-informed neural networks (PINN), both on synthetic data generated from known target parameters and on real public Danish time series data collected during the COVID-19 pandemic in Denmark. Both methods were able to estimate the target parameters, while PIR showed to perform noticeably better, especially on a compartment model with higher complexity. Given the difference in computational speed, it is concluded that the PIR method is superior to PINN for the models considered. It is also demonstrated how PIR can be applied to estimate the time-varying parameters of a compartment model that is fitted using real Danish data from the COVID-19 pandemic obtained during a period from 2020 to 2021. The study shows how data-driven and physics-informed techniques may support reliable and fast -- possibly real-time -- parameter estimation in parameter-linear nonlinear dynamic models.