Controlled Koopman Machine for Interpretable Longitudinal Personalized Alzheimer's Disease Forecasting

Effective longitudinal forecasting of cognitive decline requires more than extrapolating previous cognitive scores: it should also aim to infer the future trajectory of cognition. Nevertheless, performing longitudinal future forecasting using existing heterogeneous biological data, accommodating irregular and incomplete observations, and generalizing across cohorts is a challenging task in both biological sciences and machine learning. In this paper, we introduce the Neural Koopman Machine (NKM), a multimodal forecaster that integrates group-wise biological encoders, subject-conditioned control, and a spectrally normalized shared Koopman transition. NKM explicitly separates stable population-level dynamics from individualized, observation-driven forecasting while omitting past cognitive-history inputs. In subject-level five-fold evaluation on 949 participants from Alzheimer's Disease Neuroimaging Initiative (ADNI), NKM achieves a macro Pearson correlation of ${0.644\pm0.044}$ across three different cognitive score metrics (CDRSB, MMSE, and ADAS13), exceeding competitive multi-target baselines in every fold within the experimental setup (paired fold bootstrap $\Delta r=+0.048$, 95\% percentile CI $[0.034,0.061]$). Further, when evaluated in zero-shot transfer from ADNI to 141 participants from the Australian Imaging, Biomarkers and Lifestyle (AIBL), NKM attains the highest mean MMSE correlation among the benchmarked general forecasters (${0.392\pm0.092}$). A matched latent process-noise ablation finds a small, metric-dependent benefit from mild Gaussian diffusion ($\sigma=0.05$: $r=0.636\pm0.045$ versus $0.630\pm0.043$ for its independently trained deterministic reference), rather than deterministic superiority. In-depth analysis of the learned operators reveals rapidly decaying autonomous dynamics, while ablations identify the subject-conditioned control mechanism as the model's most influential component. Together, these results show that controlled contractive dynamics provide an effective and interpretable inductive bias for multimodal cognitive forecasting.