Periodic Reporting for period 1 - ALPS (AI-based Learning for Physical Simulation)
Période du rapport: 2022-09-01 au 2025-02-28
Computer simulations are widely used in both scientific research and engineering today. While access to large datasets has led to the growing use of Artificial Intelligence (AI) and Machine Learning (ML) to enhance these simulations, there are some challenges with purely data-driven models. These models often struggle to make accurate predictions in new, unseen situations (a concept known as generalization) and are often seen as “black boxes,” meaning their inner workings are hard to understand. This lack of transparency can be a barrier to scientific progress and makes it difficult for researchers to assess their reliability and safety, which is especially important in fields involving ethical concerns.
To address these issues, we are developing a hybrid approach that blends ML techniques with traditional mathematical modeling. This will help improve the model’s ability to generalize, especially when there is limited data available, while also ensuring that the models remain interpretable by humans. A key feature of our approach is the development of algorithms that can derive simple mathematical equations from data. These general-purpose methods and algorithms will be applied to various fields, including robotics, fluid mechanics, and the modeling of complex materials.
To address these issues, we are developing a hybrid approach that blends ML techniques with traditional mathematical modeling. This will help improve the model’s ability to generalize, especially when there is limited data available, while also ensuring that the models remain interpretable by humans. A key feature of our approach is the development of algorithms that can derive simple mathematical equations from data. These general-purpose methods and algorithms will be applied to various fields, including robotics, fluid mechanics, and the modeling of complex materials.
In the first part of the project, we have introduced a new framework that uses a discrete mathematical theory and evolutionary algorithms to derive models starting from data. The mathematical theory, called discrete exterior calculus, provides a set of concepts that can be used as building blocks to model a wide range of physical phenomena. The evolutionary algorithm searches the space of possible models by combining the building blocks according to specific rules (like the grammar of a language) and producing a population of many candidate equations. The models evolve through “mutation”, “mating” and “selection of the fittest”, replicating natural processes. We have implemented this framework in an open-source software library and demonstrated that it is able to recover equations of many physical systems that state-of-the-art symbolic regression algorithms struggle to discover. We are currently improving the effectiveness of our algorithms using modern high-performance computing techniques and developing smarter search strategies. We have also started applying our methods in several fields including the development of new accurate and interpretable models for simulating fluid and traffic flows.
Our findings could have a significant impact across many fields, including economics, where the demand for clear and understandable knowledge is increasing. By integrating humans into the modeling process using AI and ML, researchers and engineers can gain valuable insights into how complex systems work. Additionally, combining mathematical symbols with AI may lead to a paradigm shift in the field towards neuro-symbolic models—hybrid models that could offer better generalization than current methods. Further investigation is needed to explore the potential and effectiveness of these neuro-symbolic models. Ultimately, as a part of the high gains of the project, we expect our approach to contribute to the development of a safe and transparent AI.