---
title: "Physics-informed Spline Learning for Nonlinear Dynamics Discovery"
authors:
  - "Fangzheng Sun"
  - "Yang Liu"
  - "Hao Sun"
doi: "10.24963/ijcai.2021/283"
url: "https://ijcai.org/proceedings/2021/283"
pdf: "https://ijcai.org/proceedings/2021/0283.pdf"
venue: "Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence"
year: 2021
pages: "2054-2061"
track: "Main Track"
keywords:
  - "Machine Learning: Learning Sparse Models"
  - "Machine Learning: Explainable/Interpretable Machine Learning"
  - "Knowledge Representation and Reasoning: Leveraging Knowledge and Learning"
---

# Physics-informed Spline Learning for Nonlinear Dynamics Discovery

Fangzheng Sun, Yang Liu, Hao Sun

*Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence*, pages 2054-2061. DOI: [10.24963/ijcai.2021/283](https://doi.org/10.24963/ijcai.2021/283)

## Abstract

Dynamical systems are typically governed by a set of linear/nonlinear differential equations. Distilling the analytical form of these equations from very limited data remains intractable in many disciplines such as physics, biology, climate science, engineering and social science. To address this fundamental challenge, we propose a novel Physics-informed Spline Learning (PiSL) framework to discover parsimonious governing equations for nonlinear dynamics, based on sparsely sampled noisy data. The key concept is to (1) leverage splines to interpolate locally the dynamics, perform analytical differentiation and build the library of candidate terms, (2) employ sparse representation of the governing equations, and (3) use the physics residual in turn to inform the spline learning. The synergy between splines and discovered underlying physics leads to the robust capacity of dealing with high-level data scarcity and noise. A hybrid sparsity-promoting alternating direction optimization strategy is developed for systematically pruning the sparse coefficients that form the structure and explicit expression of the governing equations. The efficacy and superiority of the proposed method have been demonstrated by multiple well-known nonlinear dynamical systems, in comparison with two state-of-the-art methods.