Abstract—We address the problem of reducing a task cost functional W(S), defined over Sobolev-class signals S, when W is invariant under a global symmetry group G ⊂ Diff(M) and only accessible as a black-box. Such settings arise in machine learning, imaging, and inverse problems, where performance metrics are non-differentiable and internal to pretrained models. We propose a variational method that leverages symmetry to construct explicit, symmetry-breaking deformations of the input. By minimizing an auxiliary energy, we obtain a gauge field whose induced deformation h = A_ϕ[S] lies generically transverse to the G-orbit. We show that—even for discontinuous W—a simple double-sign test on h descends to a strictly lower-cost region with positive probability, and almost surely under mild geometric conditions. This method requires no model gradients or labels and operates entirely at test time. It offers a principled mechanism for optimizing invariant cost functionals via Lie-algebraic flows, with applications to black-box systems and symmetry-constrained tasks.

Keywords—Symmetry breaking, non-smooth optimisation, machine learning

Cite: Mikhail Osipov, "Symmetry-Breaking Lie Flows for Optimisation of Discontinuous Functionals," International Journal of Applied Physics and Mathematics, vol. 15, no. 2, pp. 91-101, 2025.

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