Polynomial approximations are a powerful tool across machine learning, uncertainty quantification and design optimisation. They allow for complex functions to be approximated with simpler, know functions, with quantifiable error. In this project Turing researchers, together with other university partners, are leading the development of a set of open source utilities called Effective Quadratures, which construct and utilise polynomial approximations for multi-dimensional data-centric engineering problems.
Explaining the science
Researchers in this project are combining classical strategies for constructing polynomials with more recent ideas (based on compressive sensing and multivariate least squares).
This combination allows the researchers to evaluate either a model or work with an existing dataset. The resulting input-output data pairs are fed into a polynomial model, where the polynomial basis is chosen based on deduced information about the inputs. Once the polynomial coefficients are computed, it's possible to compute statistics and sensitivity metrics, and find 'subspaces' in the dataset which can reduce its size and complexity. This makes a polynomial model a powerful tool for machine learning compared to computational models.
There are multiple research questions that this project is attempting to solve. These include:
- Developing near optimal quadrature rules for high-dimensional numerical integration.
- Developing strategies for supervised machine learning with reduced data.
- Deploying easy-to-use utilities in industrial design workflows for propagating uncertainties in computer models.
This work is being applied to the aerospace and mechanical engineering sectors, where approximations are needed to deal with the multi-dimensional problems that arise in these fields.