# The Signature Kernel is the solution of a Goursat PDE

arXiv: https://arxiv.org/abs/2006.14794 · HTML: https://mlsanity.com/paper/2006.14794/

## Key facts

- Cited by 58 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Jun 2020.

## Built on by (most cited first, 30 of 58)

- [Theoretical Foundations of Deep Selective State-Space Models](https://mlsanity.com/paper/2402.19047/) (2024)
- [Efficient and Accurate Gradients for Neural SDEs](https://mlsanity.com/paper/2105.13493/) (2021)
- [Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamics](https://mlsanity.com/paper/2110.10249/) (2021)
- [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)
- [Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes](https://mlsanity.com/paper/2109.03582/) (2021)
- [Neural signature kernels as infinite-width-depth-limits of controlled ResNets](https://mlsanity.com/paper/2303.17671/) (2023)
- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Global universal approximation of functional input maps on weighted spaces](https://mlsanity.com/paper/2306.03303/) (2023)
- [Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes](https://mlsanity.com/paper/2402.18477/) (2024)
- [Structured Linear CDEs: Maximally Expressive and Parallel-in-Time Sequence Models](https://mlsanity.com/paper/2505.17761/) (2025)
- [Approximate Bayesian Computation with Path Signatures](https://mlsanity.com/paper/2106.12555/) (2021)
- [New directions in the applications of rough path theory](https://mlsanity.com/paper/2302.04586/) (2023)
- [Random Fourier Signature Features](https://mlsanity.com/paper/2311.12214/) (2023)
- [Exact Gradients for Stochastic Spiking Neural Networks Driven by Rough Signals](https://mlsanity.com/paper/2405.13587/) (2024)
- [Amortised Likelihood-free Inference for Expensive Time-series Simulators with Signatured Ratio Estimation](https://mlsanity.com/paper/2202.11585/) (2022)
- [Reservoir kernels and Volterra series](https://mlsanity.com/paper/2212.14641/) (2022)
- [The Signature Kernel](https://mlsanity.com/paper/2305.04625/) (2023)
- [Free-Form Variational Inference for Gaussian Process State-Space Models](https://mlsanity.com/paper/2302.09921/) (2023)
- [SigFormer: Signature Transformers for Deep Hedging](https://mlsanity.com/paper/2310.13369/) (2023)
- [Koopman-Equivariant Gaussian Processes](https://mlsanity.com/paper/2502.06645/) (2025)
- [pySigLib -- Fast Signature-Based Computations on CPU and GPU](https://mlsanity.com/paper/2509.10613/) (2025)
- [The Gaussian kernel on the circle and spaces that admit isometric embeddings of the circle](https://mlsanity.com/paper/2302.10623/) (2023)
- [The Volterra signature](https://mlsanity.com/paper/2603.04525/) (2026)
- [Efficient Training of Neural Stochastic Differential Equations by Matching Finite Dimensional Distributions](https://mlsanity.com/paper/2410.03973/) (2024)
- [ParallelFlow: Parallelizing Linear Transformers via Flow Discretization](https://mlsanity.com/paper/2504.00492/) (2025)
- [Gaussian kernels on non-simply-connected closed Riemannian manifolds are never positive definite](https://mlsanity.com/paper/2303.06558/) (2023)
- [Path Signatures for Diversity in Probabilistic Trajectory Optimisation](https://mlsanity.com/paper/2308.04071/) (2023)
- [High Rank Path Development: an approach of learning the filtration of stochastic processes](https://mlsanity.com/paper/2405.14913/) (2024)
- [Numerical Schemes for Signature Kernels](https://mlsanity.com/paper/2502.08470/) (2025)

## Abstract (arXiv)

Recently, there has been an increased interest in the development of kernel methods for learning with sequential data. The signature kernel is a learning tool with potential to handle irregularly sampled, multivariate time series. In "Kernels for sequentially ordered data" the authors introduced a kernel trick for the truncated version of this kernel avoiding the exponential complexity that would have been involved in a direct computation. Here we show that for continuously differentiable paths, the signature kernel solves a hyperbolic PDE and recognize the connection with a well known class of differential equations known in the literature as Goursat problems. This Goursat PDE only depends on the increments of the input sequences, does not require the explicit computation of signatures and can be solved efficiently using state-of-the-arthyperbolic PDE numerical solvers, giving a kernel trick for the untruncated signature kernel, with the same raw complexity as the method from "Kernels for sequentially ordered data", but with the advantage that the PDE numerical scheme is well suited for GPU parallelization, which effectively reduces the complexity by a full order of magnitude in the length of the input sequences. In addition, we extend the previous analysis to the space of geometric rough paths and establish, using classical results from rough path theory, that the rough version of the signature kernel solves a rough integral equation analogous to the aforementioned Goursat PDE. Finally, we empirically demonstrate the effectiveness of our PDE kernel as a machine learning tool in various machine learning applications dealing with sequential data. We release the library sigkernel publicly available at https://github.com/crispitagorico/sigkernel.