DSSMs: State Space Models with Explicit Memory via Delay Differential Equations

• URL: http://arxiv.org/abs/2607.10244v1
• Date: Sat, 11 Jul 2026 10:19:21 GMT
• Title: DSSMs: State Space Models with Explicit Memory via Delay Differential Equations
• Abstract summary: State Space Models (SSMs) have emerged as a powerful paradigm for efficient long-sequence modeling.<n>We propose Delay State Space Models (DSSMs), a delay differential equation (DDE)-inspired extension of diagonal SSMs.<n>We show that DSSMs substantially improve targeted delayed-retrieval tasks while outperforming S4D on most standard sequence metrics.
• Score: 9.12852957469023
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: State Space Models (SSMs) have emerged as a powerful paradigm for efficient long-sequence modeling, offering parallel training and fast linear-time recurrent inference. However, like other recurrent architectures, SSMs must compress an unbounded history into a fixed-size state, which limits context retention and makes precise retrieval over long-range context inherently difficult. To overcome this limitation, we propose Delay State Space Models (DSSMs), a delay differential equation (DDE)-inspired extension of diagonal SSMs that augments discrete SSM recurrences with explicit delayed-state feedback. Making explicit delayed feedback practical requires new stability parameterization, history management, and FFT-training tools. We address these challenges with a practical discretization and parameterization grounded in a simple delay-independent stability condition. To bypass direct time-domain kernel construction, we derive the DSSM transfer function and compute kernels in the frequency domain, using a kernel contour shift to suppress aliasing and recover accurate FFT training. Empirically, DSSMs substantially improve targeted delayed-retrieval tasks while outperforming S4D on most standard sequence metrics and remaining close on the others.

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