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

• URL: http://arxiv.org/abs/2605.25826v1
• Date: Mon, 25 May 2026 13:22:07 GMT
• Title: Branched Signature Kernel Solvers for ODEs with rough Single-Trajectory signals
• Abstract summary: We develop a branched signature kernel solver for ordinary differential equations driven by a single trajectory.<n>A emphcount-sampling construction turns the single observation into a hierarchical family of $N+1$ nested training paths.<n>A kernel-collocation framework places the ansatz either on the highest-order derivative of the solution or on the solution itself.
• Score: 2.94985644773306
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• Explicit block-encodings for biharmonic boundary-value problems [1.5922179327791923]
  We develop quantum linear-system algorithms for periodic, simply supported, and Dirichlet--Neumann boundary conditions.<n>For Dirichlet--Neumann problems, we introduce a second-order boundary-corrected finite-difference discretization.<n>We also formulate a coupled-Laplace system with additional boundary unknowns.
  arXiv  Detail & Related papers  (2026-07-24T15:20:32Z)
• Neural Discovery of Memory and Nonlocal Kernels in Integro-Differential Equations with Constrained Kolmogorov--Arnold Networks [0.0]
  Existing identification methods often rely on problem-specific analytical derivations, specialized observation requirements, or restrictive assumptions about the kernel.<n>We propose a differentiable-solver-based framework for discovering memory and nonlocal kernels directly fromtemporal observations.
  arXiv  Detail & Related papers  (2026-07-13T05:40:37Z)
• Effective Covariance Dynamics in Solvable High-Dimensional GANs [2.639628765667673]
  We study a solvable high-dimensional model of generative adversarial network (GAN) training in which a linear generator learns a low-dimensional subspace from data.<n>We prove that the microscopic training process converges, in the high-dimensional limit, to deterministic ordinary differential equations governed by this effective covariance.
  arXiv  Detail & Related papers  (2026-06-25T16:33:54Z)
• Closed-Form Spectral Regularization for Multi-Task Model Merging [96.82449201305234]
  Model merging combines several independently fine-tuned experts into a single multi-task model without any training data.<n>State-of-the-art merging methods formulate merging as a layer-wise interference problem.<n>We propose SWUDI, a closed-form method that combines a soft exponential filter, which matches the gradient-flow trajectory of iterative descent.
  arXiv  Detail & Related papers  (2026-06-05T14:00:47Z)
• Quantum Differential Equation Solver via Hybrid Oscillator-Qubit Linear Combination of Hamiltonian Simulations [7.438217717051814]
  We introduce a hybrid oscillator-qubit formulation of linear combination of Hamiltonian simulation (LCHS)<n>Instead of representing the quadrature rule with a discrete-variable (DV) ancilla register in qubit-only LCHS, the method encodes the LCHS kernel in a continuous-variable ancillary mode.<n>In the heat-equation benchmarks, the Law--Eberly protocol achieves end-to-end solution fidelity at least 99.90%.
  arXiv  Detail & Related papers  (2026-05-11T15:20:35Z)
• Fidelity-informed neural pulse compilation of a continuous family of quantum gates with uncertainty-margin analysis [0.0]
  We develop a neural pulse-compilation framework for a family of single-qubit gates on a three-qubit liquid-state nuclear magnetic resonance (NMR) processor.<n>Instead of decomposing each target unitary into a sequence of calibrated basis gates, the method learns a direct map from the axis-angle parameters of an arbitrary U in SU(2) operation.<n>We show numerically that a single model generalizes across a continuous range of gate parameters and experimentally validate representative compiled pulses on a benchtop three-qubit NMR device.
  arXiv  Detail & Related papers  (2026-04-13T11:17:18Z)
• On the Mechanism and Dynamics of Modular Addition: Fourier Features, Lottery Ticket, and Grokking [49.1352577985191]
  We present a comprehensive analysis of how two-layer neural networks learn features to solve the modular addition task.<n>Our work provides a full mechanistic interpretation of the learned model and a theoretical explanation of its training dynamics.
  arXiv  Detail & Related papers  (2026-02-18T20:25:13Z)
• Generalizing GNNs with Tokenized Mixture of Experts [75.8310720413187]
  We show that improving stability requires reducing reliance on shift-sensitive features, leaving an irreducible worst-case generalization floor.<n>We propose STEM-GNN, a pretrain-then-finetune framework with a mixture-of-experts encoder for diverse computation paths.<n>Across nine node, link, and graph benchmarks, STEM-GNN achieves a stronger three-way balance, improving robustness to degree/homophily shifts and to feature/edge corruptions while remaining competitive on clean graphs.
  arXiv  Detail & Related papers  (2026-02-09T22:48:30Z)
• Variational Bayesian Flow Network for Graph Generation [54.94088904387278]
  We propose Variational Bayesian Flow Network (VBFN) for graph generation.<n>VBFN performs variational lifting to a tractable joint Gaussian variational belief family governed by structured precisions.<n>On synthetic and molecular graph datasets, VBFN improves fidelity and diversity, and surpasses baseline methods.
  arXiv  Detail & Related papers  (2026-01-30T03:59:38Z)
• Lindbladian reverse engineering for general non-equilibrium steady states: A scalable null-space approach [49.1574468325115]
  We introduce a method for reconstructing the corresponding Lindbaldian master equation given any target NESS. The kernel (null-space) of the correlation matrix corresponds to Lindbladian solutions. We illustrate the method in different systems, ranging from bosonic Gaussian to dissipative-driven collective spins.
  arXiv  Detail & Related papers  (2024-08-09T19:00:18Z)
• Stable Nonconvex-Nonconcave Training via Linear Interpolation [51.668052890249726]
  This paper presents a theoretical analysis of linearahead as a principled method for stabilizing (large-scale) neural network training. We argue that instabilities in the optimization process are often caused by the nonmonotonicity of the loss landscape and show how linear can help by leveraging the theory of nonexpansive operators.
  arXiv  Detail & Related papers  (2023-10-20T12:45:12Z)
• Local Convergence of Gradient Descent-Ascent for Training Generative Adversarial Networks [20.362912591032636]
  We study the local dynamics of gradient descent-ascent (GDA) for training a GAN with a kernel-based discriminator. We show phase transitions that indicate when the system converges, oscillates, or diverges.
  arXiv  Detail & Related papers  (2023-05-14T23:23:08Z)
• Towards Understanding Generalization via Decomposing Excess Risk Dynamics [13.4379473119565]
  We analyze the generalization dynamics to derive algorithm-dependent bounds, e.g., stability. Inspired by the observation that neural networks show a slow convergence rate when fitting noise, we propose decomposing the excess risk dynamics. Under the decomposition framework, the new bound accords better with the theoretical and empirical evidence compared to the stability-based bound and uniform convergence bound.
  arXiv  Detail & Related papers  (2021-06-11T03:42:45Z)
• Incorporating NODE with Pre-trained Neural Differential Operator for Learning Dynamics [73.77459272878025]
  We propose to enhance the supervised signal in learning dynamics by pre-training a neural differential operator (NDO) NDO is pre-trained on a class of symbolic functions, and it learns the mapping between the trajectory samples of these functions to their derivatives. We provide theoretical guarantee on that the output of NDO can well approximate the ground truth derivatives by proper tuning the complexity of the library.
  arXiv  Detail & Related papers  (2021-06-08T08:04:47Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.