FracEncoder: Towards Adaptive Cognitive Trajectories via Fractional-Order Context Encoding

In real-world tasks such as robotic control and autonomous driving, agents often face non-Markovian environments due to incomplete observations. Solving such information-impoverished decision tasks requires reconstructing the context from history. However, existing recurrent encoders have two fundamental limitations: First, the model suffers from a Markovian compression bottleneck, i.e., it only processes observation history as input but neglects thought process history. This leads to the loss of deep semantic cues from early interactions, which are easily overwritten by noise over time due to recursion. Second, it lacks the adaptive capability to adjust the memory span for different tasks. To address these limitations, we propose FracEncoder, a context encoder based on fractional-order neural differential equations. First, FracEncoder leverages the nonlocal property of the fractional-order derivative and uses a power-law kernel to directly incorporate the trajectory of the thought process into the evolution of the current state, effectively overcoming the temporal locality limitations of traditional models. Second, the fractional order $\alpha$ is set as an end-to-end learnable parameter. It explicitly characterizes how strongly the system anchors to its history, so the model can discover a suitable memory span for each task. We evaluate FracEncoder on more than ten benchmarks across three settings, namely partially observable tasks, meta reinforcement learning, and delayed-observation control. FracEncoder consistently matches or outperforms representative baselines, including GPIDE, GRU-ODE, ReSeL, and LRU. On the challenging 8-step delayed-observation task, it reaches average returns of $-511.5 \pm 7.9$ on Pendulum and $732.1 \pm 152.7$ on Hopper. As a plug-in module, FracEncoder also yields stable gains across different reinforcement learning algorithms. Ablation studies further show that the learned $\alpha$ aligns with the intrinsic memory demand of each task, providing an interpretable physical measure of the cognitive complexity of non-Markovian tasks. The code is available at https://anonymous.4open.science/r/FracEncoder-60BA.