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Researchers have determined how to build reliable machine learning models that can understand complex equations in real-world situations while using far less training data than is normally expected.

It’s surprising how little data you need to end up with a reliable model Nicolas BoulléThe researchers, from the University of Cambridge and Cornell University, found that for partial differential equations - a class of physics equations that describe how things in the natural world evolve in space and time - machine learning models can produce reliable results even when they are provided with limited data.

Their results , reported in the

*Proceedings of the National Academy of Sciences*, could be useful for constructing more timeand cost-efficient machine learning models for applications such as engineering and climate modelling.

Most machine learning models require large amounts of training data before they can begin returning accurate results. Traditionally, a human will annotate a large volume of data - such as a set of The researchers found that PDEs that model diffusion have a structure that is useful for designing AI models. -Using a simple model, you might be able to enforce some of the physics that you already know into the training data set to get better accuracy and performance,- said Boullé.

The researchers constructed an efficient algorithm for predicting the solutions of PDEs under different conditions by exploiting the short and long-range interactions happening. This allowed them to build some mathematical guarantees into the model and determine exactly how much training data was required to end up with a robust model.

-It depends on the field, but for physics, we found that you can actually do a lot with a very limited amount of data,- said Boullé. -It’s surprising how little data you need to end up with a reliable model. Thanks to the mathematics of these equations, we can exploit their structure to make the models more efficient.-

The researchers say that their techniques will allow data scientists to open the -black box- of many machine learning models and design new ones that can be interpreted by humans, although future research is still needed.

-We need to make sure that models are learning the right things, but machine learning for physics is an exciting field - there are lots of interesting maths and physics questions that AI can help us answer,- said Boullé.

Nicolas Boullé, Diana Halikias, and Alex Townsend. - Elliptic PDE learning is provably data-efficient.- PNAS (2023). DOI: 10.1073/pnas.2303904120

**Reference:**Nicolas Boullé, Diana Halikias, and Alex Townsend. - Elliptic PDE learning is provably data-efficient.- PNAS (2023). DOI: 10.1073/pnas.2303904120