# A Physics-Constrained Data-Driven Approach Based on Locally Convex Reconstruction for Noisy Database

## TL;DR

This paper introduces a physics-constrained, locally convex reconstruction method called LCDD that improves accuracy and robustness in data-driven simulations, especially with noisy and high-dimensional data.

## Contribution

The paper proposes the LCDD approach combining local convexity and physics constraints, enhancing noise robustness and accuracy in data-driven computational models.

## Key findings

- LCDD achieves nearly tenfold accuracy improvement over standard methods with noisy data.
- The method maintains linear exactness for linear stress-strain relations.
- Effective in high-dimensional, sparse data scenarios in engineering applications.

## Abstract

Physics-constrained data-driven computing is an emerging hybrid approach that integrates universal physical laws with data-driven models of experimental data for scientific computing. A new data-driven simulation approach coupled with a locally convex reconstruction, termed the local convexity data-driven (LCDD) computing, is proposed to enhance accuracy and robustness against noise and outliers in data sets in the data-driven computing. In this approach, for a given state obtained by the physical simulation, the corresponding optimum experimental solution is sought by projecting the state onto the associated local convex manifold reconstructed based on the nearest experimental data. This learning process of local data structure is less sensitive to noisy data and consequently yields better accuracy. A penalty relaxation is also introduced to recast the local learning solver in the context of non-negative least squares that can be solved effectively. The reproducing kernel approximation with stabilized nodal integration is employed for the solution of the physical manifold to allow reduced stress-strain data at the discrete points for enhanced effectiveness in the LCDD learning solver. Due to the inherent manifold learning properties, LCDD performs well for high-dimensional data sets that are relatively sparse in real-world engineering applications. Numerical tests demonstrated that LCDD enhances nearly one order of accuracy compared to the standard distance-minimization data-driven scheme when dealing with noisy database, and a linear exactness is achieved when local stress-strain relation is linear.

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Source: https://tomesphere.com/paper/1907.12651