ASP-Assisted Symbolic Regression: Uncovering Hidden Physics in Fluid Mechanics

Unlike conventional Machine-Learning (ML) approaches, often criticized as "black boxes", Symbolic Regression (SR) stands out as a powerful tool for revealing interpretable mathematical relationships in complex physical systems, requiring no a priori assumptions about models' structures. Motivated by the recognition that, in fluid mechanics, an understanding of the underlying flow physics is as crucial as accurate prediction, this study applies SR to model a fundamental three-dimensional (3D) incompressible flow in a rectangular channel, focusing on the (axial) velocity and pressure fields under laminar conditions. By employing the PySR library, compact symbolic equations were derived directly from numerical simulation data, revealing key characteristics of the flow dynamics. These equations not only approximate the parabolic velocity profile and pressure drop observed in the studied fluid flow, but also perfectly coincide with analytical solutions from the literature. Furthermore, we propose an innovative approach that integrates SR with the knowledge-representation framework of Answer Set Programming (ASP), combining the generative power of SR with the declarative reasoning strengths of ASP. The proposed hybrid SR/ASP framework ensures that the SR-generated symbolic expressions are not only statistically accurate, but also physically plausible, adhering to domain-specific principles. Overall, the study highlights two key contributions: SR's ability to simplify complex flow behaviours into concise, interpretable equations, and the potential of knowledge-representation approaches to improve the reliability and alignment of data-driven SR models with domain principles. Insights from the examined 3D channel flow pave the way for integrating such hybrid approaches into efficient frameworks, [...] where explainable predictions and real-time data analysis are crucial.

Machine Learning as Iterated Belief Change a la Darwiche and Pearl

Artificial Neural Networks (ANNs) are powerful machine-learning models capable of capturing intricate non-linear relationships. They are widely used nowadays across numerous scientific and engineering domains, driving advancements in both research and real-world applications. In our recent work, we focused on the statics and dynamics of a particular subclass of ANNs, which we refer to as binary ANNs. A binary ANN is a feed-forward network in which both inputs and outputs are restricted to binary values, making it particularly suitable for a variety of practical use cases. Our previous study approached binary ANNs through the lens of belief-change theory, specifically the Alchourron, Gardenfors and Makinson (AGM) framework, yielding several key insights. Most notably, we demonstrated that the knowledge embodied in a binary ANN (expressed through its input-output behaviour) can be symbolically represented using a propositional logic language. Moreover, the process of modifying a belief set (through revision or contraction) was mapped onto a gradual transition through a series of intermediate belief sets. Analogously, the training of binary ANNs was conceptualized as a sequence of such belief-set transitions, which we showed can be formalized using full-meet AGM-style belief change. In the present article, we extend this line of investigation by addressing some critical limitations of our previous study. Specifically, we show that Dalal's method for belief change naturally induces a structured, gradual evolution of states of belief. More importantly, given the known shortcomings of full-meet belief change, we demonstrate that the training dynamics of binary ANNs can be more effectively modelled using robust AGM-style change operations -- namely, lexicographic revision and moderate contraction -- that align with the Darwiche-Pearl framework for iterated belief change.