Branched Signature Kernel Solvers for ODEs with rough Single-Trajectory signals
2026-05-25 • Computational Engineering, Finance, and Science
Computational Engineering, Finance, and ScienceMachine Learning
AI summaryⓘ
The authors create a new method to solve differential equations when you only have one observed example of the input signal, which is common in fields like earthquake engineering and finance. They introduce a way to generate multiple training paths from this single observation using a "count-sampling" approach, enabling existing tools designed for many examples to work here. They also use a kernel-based framework to approximate solutions either by focusing on derivatives or the solution itself. Their method is proven to approximate a wide range of functions and can update predictions online. Tests on various complex problems show their solver is accurate and stable.
branched signature kernelordinary differential equationscount-samplingkernel collocationHairer-Kelly morphismgeometric signaturestreaming learningNewton methodrough signalsuniversal approximation theorem
Authors
Munawar Ali, Qi Feng, Charlie Pyle, George Xu
Abstract
We develop a branched signature kernel solver for linear and nonlinear ordinary differential equations driven by a \emph{single observed trajectory} of a possibly rough forcing signal -- a setting that arises naturally in earthquake engineering, finance, biology, and structural health monitoring, where the forcing is observed exactly once and the solver must respect the underlying physical law without recourse to an ensemble of realizations. Two ingredients are new. First, a \emph{count-sampling} construction turns the single observation into a hierarchical family of $N+1$ nested training paths on which the branched signature kernel can be evaluated; this allows the signature kernel machinery, originally designed for multi-realization regression problems, to operate on a single-trajectory observation. Second, a kernel-collocation framework places the ansatz either on the highest-order derivative of the solution (with lower derivatives recovered by integrating the kernel) or on the solution itself (after $m$-fold integration of the ODE). We prove a universal approximation theorem for the branched signature kernel, leveraging the Hairer--Kelly morphism to express branched signature evaluations through geometric signatures of time-extended paths. The offline solver is extended to a streaming Test/Train/Retrain protocol with closed-form online updates in the linear case and scalar Newton steps in the nonlinear case. Numerical experiments on six benchmarks (El-Centro earthquake displacement, the Solow capital-stock model, an fBM-driven second-order ODE, a forced Duffing oscillator, a path-dependent Arias-intensity-degraded oscillator with variable coefficients, and a noisy Kuramoto phase-oscillator system) show that the branched signature-kernel solver delivers accurate, stable predictions across all regimes.