We introduce a training-efficient framework for time-series learning that combines random features with controlled differential equations (CDEs). In this approach, large randomly parameterized CDEs act as continuous-time reservoirs, mapping input paths to rich representations. Only a linear readout layer is trained, resulting in fast, scalable models with strong inductive bias. Building on this foundation, we propose two variants: (i) Random Fourier CDEs (RF-CDEs): these lift the input signal using random Fourier features prior to the dynamics, providing a kernel-free approximation of RBF-enhanced sequence models; (ii) Random Rough DEs (R-RDEs): these operate directly on rough-path inputs via a log-ODE discretization, using log-signatures to capture higher-order temporal interactions while remaining stable and efficient. We prove that in the infinite-width limit, these model induces the RBF-lifted signature kernel and the rough signature kernel, respectively, offering a unified perspective on random-feature reservoirs, continuous-time deep architectures, and path-signature theory. We evaluate both models across a range of time-series benchmarks, demonstrating competitive or state-of-the-art performance. These methods provide a practical alternative to explicit signature computations, retaining their inductive bias while benefiting from the efficiency of random features.

翻译：我们提出了一种结合随机特征与控制微分方程(CDEs)的高效时间序列学习框架。该方法采用大规模随机参数化的CDE作为连续时间储备池，将输入路径映射为丰富的表示。仅需训练线性读出层，即可获得具有强归纳偏置的快速可扩展模型。基于此框架，我们提出两种变体：(i) 随机傅里叶CDE(RF-CDEs)：在动力学系统前通过随机傅里叶特征对输入信号进行升维，提供RBF增强序列模型的无核近似；(ii) 随机粗糙微分方程(R-RDEs)：通过log-ODE离散化直接处理粗糙路径输入，利用对数签名捕捉高阶时间交互作用，同时保持稳定性和计算效率。我们证明在无限宽度极限下，这两种模型分别诱导出RBF升维签名核与粗糙签名核，为随机特征储备池、连续时间深度架构和路径签名理论提供了统一视角。我们在多个时间序列基准测试中评估这两种模型，结果表明其性能达到竞争水平或最优水平。这些方法为显式签名计算提供了实用替代方案，在保留归纳偏置的同时受益于随机特征的效率优势。