PCD: Predictive Coding Dynamics for Irregularly Sampled Time Series

Tianrui Gou; Jiayangmei Xu; Xingzhou Zhang; Lei Sha

cs.LGOct 30, 2026

PCD: Predictive Coding Dynamics for Irregularly Sampled Time Series

Authors: Tianrui Gou, Jiayangmei Xu, Xingzhou Zhang, Lei Sha

Organizations: School of Artificial Intelligence, Beihang University, Beijing 100191, China. · Department of Foreign Languages, Beihang University, Beijing 100191, China.

Abstract

Dynamic systems evolve continuously over time, whereas observations in real scenarios are often sparse, irregular, and asynchronous. Accurate prediction therefore requires inferring latent state evolution between observations, as future values depend on partially observed trajectories. We propose Predictive Coding Dynamics (PCD), a residual feedback mechanism for irregular continuous dynamic models. PCD maintains previous reconstructions, predicts observed entries from them, and incorporates masked signed residuals into state recognition through an energy penalty. The mechanism is implemented with ODE-RNN and Latent ODE, supporting event-based hidden updates and latent state recognition. Experiments on forecasting and trajectory recovery tasks show that PCD improves continuous dynamic backbones under sparse and irregular sampling. The results further indicate that predictive error feedback improves latent state estimation when observations constrain the current trajectory. These results suggest that PCD provides a residual feedback framework that jointly supports trajectory generation, observation consistency checking, and state refinement in continuous-time series modeling. The source code is available at https://anonymous.4open.science/r/PCD-for-paper-92F1/.

Figures & tables

Fig. 1: Predictive Coding Dynamics (PCD) feedback loop for irregular continuous-time modeling. The backbone reconstructs a trajectory, the PC predictor generates predictions from the cached reconstruction, and weighted masked residuals are incorporated into state recognition for the next reconstruction.

Dataset	Domain	Observation Type	Task
MuJoCo [ 48 , 49 ]	Physical simulation	Multivariate trajectories	Forecasting and interpolation
PhysioNet 2012 [ 50 ]	Clinical monitoring	Irregular ICU measurements	Forecasting and interpolation
USHCN [ 52 ]	Climate monitoring	Irregular temperature observations	Forecasting
Human Activity [ 51 ]	Human sensing	Irregular sensor observations	Forecasting
Darts [ 53 ]	Time series benchmark	Multiple regular time series	Forecasting

TABLE I: Datasets and Evaluation Tasks

Method	Regular	30% Dropped	50% Dropped	70% Dropped
GRU- $\Delta t$ [ 54 ]	$0.223\pm 0.020$	$0.198\pm 0.036$	$0.193\pm 0.015$	$0.196\pm 0.028$
GRU-D [ 1 ]	$0.578\pm 0.042$	$0.608\pm 0.032$	$0.587\pm 0.039$	$0.579\pm 0.052$
GRU-ODE [ 5 ]	$0.856\pm 0.016$	$0.857\pm 0.015$	$0.852\pm 0.015$	$0.861\pm 0.015$
ODE-RNN [ 4 ]	$0.328\pm 0.225$	$0.274\pm 0.213$	$0.237\pm 0.110$	$0.267\pm 0.217$
Neural CDE [ 6 ]	$0.028\pm 0.002$	$0.027\pm 0.000$	$0.027\pm 0.001$	$0.026\pm 0.001$
ANCDE [ 55 ]	$0.026\pm 0.001$	$0.025\pm 0.001$	$0.025\pm 0.001$	$0.024\pm 0.001$

TABLE II: MuJoCo Forecasting with Increasingly Sparse Histories

	USHCN		Human Activity
Method	MSE ( $\times 10^{-1}$ )	MAE ( $\times 10^{-1}$ )	MSE ( $\times 10^{-3}$ )	MAE ( $\times 10^{-2}$ )
GRU-D [ 1 ]	$5.54\pm 0.38$	$3.40\pm 0.28$	$2.94\pm 0.05$	$3.51\pm 0.06$
CRU [ 9 ]	$6.09\pm 0.17$	$3.54\pm 0.18$	$6.97\pm 0.78$	$6.30\pm 0.47$
Crossformer [ 59 ]	$5.25\pm 0.04$	$3.27\pm 0.09$	$4.29\pm 0.20$	$4.89\pm 0.17$
Latent ODE [ 4 ]	$5.62\pm 0.03$	$3.60\pm 0.12$	$3.34\pm 0.11$	$3.94\pm 0.12$
DLinear [ 38 ]	$6.21\pm 0.00$	$3.88\pm 0.02$	$4.03\pm 0.01$	$4.21\pm 0.01$

TABLE III: Irregular Forecasting on USHCN and Human Activity

Table 5

	MSE ( $\times 10^{-2}$ )
Method	10% Observed	20% Observed	30% Observed	50% Observed
RNN $\Delta t$ [ 4 ]	$2.454$	$1.714$	$1.250$	$0.785$
RNN GRU-D [ 1 ]	$1.968$	$1.421$	$1.134$	$0.748$
ODE-RNN [ 4 ]	$1.647$	$1.209$	$0.986$	$0.665$
ODE-RNN + STEER [ 68 ]	$1.528$	$1.090$	$0.796$	$0.603$
RNN-VAE [ 4 ]	$6.514$	$6.408$	$6.305$	$6.100$

TABLE VI: MuJoCo Trajectory Interpolation at Different Observation Rates

Fig. 2: Visualization of predictive coding refinement on a synthetic one-dimensional trajectory. (a) Ground truth, Latent ODE baseline, and PCD-refined trajectories, with observations and a local zoom. (b) Absolute prediction errors of the baseline and PCD. PCD reduces the prediction deviation and follows the reference dynamics more closely.

	Forecasting MSE			Reconstruction MSE ( $\times 10^{-2}$ )
Variant	0% Dropped	50% Dropped	70% Dropped	10% Observed	30% Observed
ODE-RNN Backbone	$0.3280\pm 0.0220$	$0.2370\pm 0.0180$	$0.2670\pm 0.0250$	$1.647\pm 0.061$	$0.986\pm 0.038$
w/o Error Feedback	$0.2860\pm 0.0190$	$0.2210\pm 0.0170$	$0.2490\pm 0.0220$	$1.582\pm 0.056$	$0.951\pm 0.034$
Error Feedback Only	$0.0118\pm 0.0014$	$0.0189\pm 0.0021$	$0.0295\pm 0.0038$	$1.284\pm 0.045$	$0.872\pm 0.030$
w/o Cache	$0.0096\pm 0.0011$	$0.0148\pm 0.0017$	$0.0224\pm 0.0030$	$1.226\pm 0.042$	$0.849\pm 0.027$
w/o Energy	$0.0074\pm 0.0009$	$0.0102\pm 0.0013$	$0.0157\pm 0.0025$	$1.168\pm 0.038$	$0.823\pm 0.025$

TABLE VII: Core Component Ablation on MuJoCo with the ODE-RNN Backbone

Panel of Figure 9 — Fig. 3: Trajectory refinement under sparse observations. Snapshots compare the ground truth, backbone predictions, and successive refinement configurations. Full PCD produces states closer to the ground truth than the corresponding backbone predictions in the illustrated examples.

Table 10

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Aug 25, 2026J/K move · Enter open · S save

Single State Update Predictive Coding training for Time Series Forecasting and Anomaly Detection

Matteo Cardoni, Sam Leroux

IDLab, Department of Information and Technology, Ghent University—imec, 9052 Ghent, Belgium

Predictive Coding Online Anomaly Detection Time-Series Anomaly Detection Time Series Forecasting Online Continual Learning

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Predictive Coding (PC) is a neural learning paradigm that enables parallelizable neural network layer updates. However, the main bottleneck of PC Networks (PCN) is the sequential backwards error propagation. To tackle this, we introduce a training technique that pairs a Generative PCN with a support Encoding PCN. The two PCNs are trained in parallel to match their neural activations, without sequential propagation. We apply this to time series anomaly detection and show that our approach results in more stable, continuous, online learning.

Abstract

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