In many contexts like climate science and engineering just to cite a few, the laws of Physics can be understood as a set of constraints to be satisfied by any statistical model. The goal of this project is to lay the theoretical and methodological foundations for adaptive embedding of information stemming from natural sciences into machine learning protocols. The resulting mathematical framework can potentially be extended to tackle general highly-structured a priori knowledge available in different forms.

Explaining the science

A correctly parameterised physical model, constructed on the basis of fundamental laws, will typically demonstrate superior predictive performance to standard machine learning models when both are trained on small datasets. However, for many complex systems a fully mechanistic description may not be available, motivating the use of completely data driven approaches. Even in an abundance of data, these techniques often still perform poorly when the feature space exhibited by the observed process does not provide sufficient information to grasp the underlying complexity of the true data generating process. This is particularly the case for systems with unobserved dynamical phenomena interacting across different scales. In these scenarios, it becomes essential to increase the data efficiency by incorporating prior knowledge of physical laws. This motivates a class of hybrid models, able to incorporate aspects of both fully mechanistic and completely data-driven extremes.

Challenge aims

Despite the ubiquity of general hybrid modelling, there is complete lack of a formal framework for encoding mechanistic knowledge. Instead the field is characterised by ad-hoc methods [Bohlin, 1994, Glassey and von Stosch, 2018] that lead to an ever increasing toolbox of possible techniques for the practitioner, with few guidelines and no formality. Consequently, there is an absence of theoretical guarantees for achieving increased data efficiency and improved generalisation, or quantitative criteria to select those components of the mechanistic model which are important to preserve during embedding. The existing hybrid modelling approaches are only valid under limited regimes, have simple mechanistic components and make unrealistic assumptions in order to achieve physics/algorithm consistency. Φ-ML aims to extend the formal theoretical frameworks of hybrid models in a unified perspective. Equipped with this, it possible to begin to establish theoretical guarantees, suggest improved algorithms, and qualify the physical structure preserved by these methods, benefiting both practitioners with strong prior knowledge, and automating the process for more speculative mechanism discovery.

Potential for impact

The resulting mathematical rationale can be framed in terms of a python package.


Related activities

The exploration of this novel research area will revolve around a two-weeks workshop funded in the context of the “Theory and Methods Challenge Fortnights” Turing grant scheme. In addition, a seminar series exploring the ideas of Φ-ML in the engineering context is currently being organised and its provisional agenda is available here.

Recent updates

Due to the recent impact of coronavirus, the workshop has been postponed to late Spring/early Summer 2021.



[Bohlin, 1994] Bohlin, T. (1994). Derivation of a designers guide for iterative grey box identification of nonlinear stochastic objects. International Journal of Control, 59:1505–1524.

[Glassey and von Stosch, 2018] Glassey, J. and von Stosch, M. (2018). Hybrid modelling in process industries. CRC press

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