{"id":12328,"date":"2025-03-11T16:08:52","date_gmt":"2025-03-11T16:08:52","guid":{"rendered":"https:\/\/www.blopig.com\/blog\/?p=12328"},"modified":"2025-03-11T16:28:45","modified_gmt":"2025-03-11T16:28:45","slug":"geometric-deep-learning-meets-forces-equilibrium","status":"publish","type":"post","link":"https:\/\/www.blopig.com\/blog\/2025\/03\/geometric-deep-learning-meets-forces-equilibrium\/","title":{"rendered":"Geometric Deep Learning meets Forces & Equilibrium"},"content":{"rendered":"\n\n\n\n\n

Introduction<\/h2>\n\n\n\n

Graphs provide a powerful mathematical framework for modelling complex systems, from molecular structures to social networks. In many physical and geometric problems, nodes represent particles, and edges encode interactions, often acting like springs. This perspective aligns naturally with Geometric Deep Learning, where learning algorithms leverage graph structures to capture spatial and relational patterns.<\/p>\n\n\n\n

Understanding energy functions and the forces derived from them is fundamental to modelling such systems. In physics and computational chemistry, harmonic potentials, which penalise deviations from equilibrium positions, are widely used to describe elastic networks, protein structures, and even diffusion processes. The Laplacian matrix plays a key role in these formulations, linking energy minimisation to force computations in a clean and computationally efficient way.<\/p>\n\n\n\n

By formalising these interactions using matrix notation, we gain not only a compact representation but also a foundation for more advanced techniques such as Langevin dynamics, normal mode analysis, and graph-based neural networks for physical simulations.<\/p>\n\n\n\n\n\n\n\n

Harmonic Energy<\/h2>\n\n\n\n

Let \\(G = (V, E)\\) be an undirected graph, where \\(V = {1, \\dots, N}\\) is the set of \\(N\\) nodes, each associated with a position vector \\(\\mathbf{x}_i \\in \\mathbb{R}^d\\), and \\( E \\subseteq V \\times V \\) is the set of edges connecting pairs of nodes. The structure of the graph is encoded in the adjacency matrix \\( \\mathbf{A} \\), where \\( A{ij} = 1 \\) if nodes \\( i \\) and \\( j \\) are connected, and 0 otherwise. The degree matrix \\( \\mathbf{D} \\) is a diagonal matrix where \\( D_{ii} = \\sum_{j} A_{ij} \\) gives the degree of each node. The graph Laplacian is then defined as \\( \\mathbf{L} = \\mathbf{D} – \\mathbf{A} \\), which governs energy and force computations. Let \\(\\mathbf{X}\\) as the \\(N\\times d\\) matrix whose \\(i\\) th row is \\(\\mathbf{x}_i^\\top\\): <\/p>\n\n\n

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The harmonic (spring-like) energy of this system is:<\/p>\n\n\n

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Hence, the harmonic energy can be expressed succinctly in terms of the graph Laplacian, giving us a compact way of defining the total energy of the system.<\/p>\n\n\n

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Energy to Forces<\/h3>\n\n\n\n

Using the definition of total energy as a function of the Laplacian, we may express the force on node \\(i\\) as the derivative with respect to the position of each node<\/p>\n\n\n

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For a specific pair of connected nodes \\( (i, j) \\), we expand \\( L_{ij} = D_{ij} – A_{ij} \\), where we can show that the forces on node \\( i \\) due to node \\( j \\) and vice-versa are antisymmetric.<\/p>\n\n\n

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Near Equilibrium<\/h3>\n\n\n\n

For each edge \\(((i,j)\\), the bond energy is given by<\/p>\n\n\n

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Where<\/p>\n\n\n\n