Identification of Partial-Differential-Equations-Based Models from Noisy Data via Splines

TL;DR

This paper introduces SAPDEMI, a two-stage method combining spline-based derivative estimation and Lasso for identifying PDE models from noisy data, validated through numerical and real NASA data case studies.

Contribution

The paper presents a novel two-stage approach that efficiently identifies PDE models from noisy data using splines and Lasso, with proven statistical properties and real-world validation.

Findings

Efficient computational complexity proportional to sample size.

Accurate PDE model identification demonstrated on numerical examples.

Successful application to NASA data case study.

Abstract

We propose a two-stage method called \textit{Spline Assisted Partial Differential Equation based Model Identification (SAPDEMI)} to identify partial differential equation (PDE)-based models from noisy data. In the first stage, we employ the cubic splines to estimate unobservable derivatives. The underlying PDE is based on a subset of these derivatives. This stage is computationally efficient: its computational complexity is a product of a constant with the sample size; this is the lowest possible order of computational complexity. In the second stage, we apply the Least Absolute Shrinkage and Selection Operator (Lasso) to identify the underlying PDE-based model. Statistical properties are developed, including the model identification accuracy. We validate our theory through various numerical examples and a real data case study. The case study is based on a National Aeronautics and Space Administration (NASA) data set.