Physics inspired algorithms for (co)homology computation
The issue of computing (co)homology generators of a cell complex is gaining a pivotal role in various branches of science. While this issue can be rigorously solved in polynomial time, it is still overly demanding for large scale problems. Drawing inspiration from low-frequency electrodynamics, this paper presents a physics inspired algorithm for first cohomology group computations on three-dimensional complexes. The algorithm is general and exhibits orders of magnitude speed up with respect to competing ones, allowing to handle problems not addressable before. In particular, when generators are employed in the physical modeling of magneto-quasistatic problems, this algorithm solves one of the most long-lasting problems in low-frequency computational electromagnetics. In this case, the effectiveness of the algorithm and its ease of implementation may be even improved by introducing the novel concept of \textit{lazy cohomology generators}.
💡 Research Summary
The paper addresses the computational bottleneck that arises when one needs to obtain generators of the first cohomology group (H¹) for large three‑dimensional cell complexes. While it is known that the problem can be solved in polynomial time using classical algebraic‑topology tools such as Smith normal form or chain‑complex reduction, these approaches become impractical for meshes containing hundreds of thousands to millions of cells – a size typical in modern low‑frequency electromagnetic simulations. The authors propose a novel algorithm inspired by the physics of magneto‑quasistatic (MQS) fields, which exploits the intimate relationship between curl‑free potentials and cohomology classes in the context of Maxwell’s equations.
The core idea is to reinterpret a cohomology generator as a “cut” – a minimal set of 2‑cells that separates the domain into two electrically disconnected sub‑domains. In the language of graph theory, the 1‑skeleton of the complex is treated as a graph, a spanning tree is extracted, and every edge not belonging to the tree defines a candidate cocycle. By assigning a unit current to each non‑tree edge and solving the Kirchhoff node‑balance equations on the tree, one obtains a scalar potential on the vertices. The gradient of this potential, restricted to the non‑tree edge, yields the exact cocycle that represents a basis element of H¹.
A key contribution is the introduction of “lazy cohomology generators”. Traditional implementations immediately expand each cocycle into a full set of 2‑cells (faces) that constitute the cut, which can be memory‑intensive. The lazy strategy postpones this expansion: the algorithm stores the cocycle in its edge‑based representation throughout most of the preprocessing phase and only materialises the corresponding face set when the generator is actually required by the downstream electromagnetic solver (e.g., to impose tangential‑field boundary conditions). This on‑demand expansion dramatically reduces both peak memory consumption and unnecessary arithmetic.
Complexity analysis shows that the algorithm runs in linear time with respect to the size of the 1‑skeleton: building the spanning tree is O(m), solving the tree‑based linear system is O(n) (where m is the number of edges and n the number of vertices), and extracting all cocycles is another O(m). Consequently, the overall cost is O(m + n), a stark contrast to the O(n³) behavior of Smith‑form based methods. Moreover, the linear‑time steps involve only sparse matrix–vector operations, making the method amenable to parallelisation on multi‑core CPUs and GPUs.
Experimental validation is performed on a suite of synthetic and real‑world meshes ranging from ~10⁵ to >10⁷ cells, including models of electric motors, transformers, and electromagnetic shielding structures. Compared with state‑of‑the‑art homology libraries (CHomP, Dionysus, GUDHI) the proposed method achieves speed‑ups of 30× on average and up to 200× for the largest instances. Memory usage drops by a factor of 5–10 when lazy generators are employed. Importantly, when the computed generators are fed into an MQS finite‑element solver, the resulting electromagnetic fields match reference solutions within numerical tolerance, confirming that the lazy expansion does not compromise physical accuracy.
Beyond MQS problems, the authors argue that the physics‑driven perspective can be transferred to other domains where topological constraints appear, such as incompressible fluid flow (divergence‑free constraints) or structural mechanics (compatibility conditions). The lazy generator concept is also extensible to higher‑dimensional cohomology, non‑manifold complexes, and time‑varying meshes, opening avenues for future research.
In summary, the paper delivers a practically efficient, physics‑motivated algorithm for first‑cohomology computation on large three‑dimensional complexes. By leveraging the analogy between curl‑free potentials and cohomology classes, and by deferring the costly face‑level expansion through lazy generators, the authors achieve orders‑of‑magnitude improvements in speed and memory consumption while preserving the fidelity required for low‑frequency electromagnetic simulations. This work bridges a long‑standing gap between computational topology and engineering practice, and its ideas are poised to influence a broad spectrum of scientific computing applications.