# Branched Signature Kernel Solvers for ODEs with rough Single-Trajectory signals

arXiv: https://arxiv.org/abs/2605.25826 · HTML: https://mlsanity.com/paper/2605.25826/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 67 references that are in the map.
- First posted on arXiv 25 May 2026.

## Builds on (most cited first, 20 of 20)

- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [A Primer on the Signature Method in Machine Learning](https://mlsanity.com/paper/1603.03788/) (2016)
- [Kernels for sequentially ordered data](https://mlsanity.com/paper/1601.08169/) (2016)
- [Deep Signature Transforms](https://mlsanity.com/paper/1905.08494/) (2019)
- [The Signature Kernel is the solution of a Goursat PDE](https://mlsanity.com/paper/2006.14794/) (2020)
- [Signature moments to characterize laws of stochastic processes](https://mlsanity.com/paper/1810.10971/) (2018)
- [Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU](https://mlsanity.com/paper/2001.00706/) (2020)
- [The iisignature library: efficient calculation of iterated-integral signatures and log signatures](https://mlsanity.com/paper/1802.08252/) (2018)
- [Non-adversarial training of Neural SDEs with signature kernel scores](https://mlsanity.com/paper/2305.16274/) (2023)
- [Signature Methods in Machine Learning](https://mlsanity.com/paper/2206.14674/) (2022)
- [Distribution Regression for Sequential Data](https://mlsanity.com/paper/2006.05805/) (2020)
- [Random Fourier Signature Features](https://mlsanity.com/paper/2311.12214/) (2023)
- [Deep Signature Algorithm for Multi-dimensional Path-Dependent Options](https://mlsanity.com/paper/2211.11691/) (2022)
- [pySigLib -- Fast Signature-Based Computations on CPU and GPU](https://mlsanity.com/paper/2509.10613/) (2025)
- [Deep Signature FBSDE Algorithm](https://mlsanity.com/paper/2108.10504/) (2021)
- [Universal approximation with signatures of non-geometric rough paths](https://mlsanity.com/paper/2602.05898/) (2026)
- [A Neural RDE-based model for solving path-dependent PDEs](https://mlsanity.com/paper/2306.01123/) (2023)
- [Adaptively Truncated Signature-based Logistic Regression for Semi-parametric Functional Classification](https://mlsanity.com/paper/2507.06637/) (2025)
- [Local regression on path spaces with signature metrics](https://mlsanity.com/paper/2510.16728/) (2025)

## Abstract (arXiv)

We develop a branched signature kernel solver for linear and nonlinear ordinary differential equations driven by a single observed trajectory of a possibly rough forcing signal--a setting common within earthquake engineering, finance, biology, and structural health monitoring, where only one forcing realization is available, and the solver must respect the underlying physical law without an ensemble of realizations. We first introduce a count-sampling construction method to turn 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. Then we build a kernel-collocation framework, which places the ansatz either on the highest-order derivative of the solution or on the solution itself. 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 optional closed-form online updates in both linear and nonlinear cases. Numerical experiments on six benchmarks show accurate, stable predictions across all regimes.