9 Hybrid models – Dynamics

9.1 H1. Mechanistic model with learned residual

Fit a model from Chapter 7, then fit a supervised learner to its residuals. The final prediction is the sum.

This is the most interpretable hybrid and provides a clean decomposition: the mechanistic component’s contribution and the learned correction’s contribution are separately measurable. It should be included in any comparison as the reference hybrid, because it is the specification against which more ambitious hybrids must justify their additional complexity.

Minimum design: whatever the mechanistic component requires.

9.2 H2. Learned parameterization

A supervised learner maps subject-level baseline information to the parameters of the mechanistic model:

\[ (\hat{k}_i, \hat{\tau}_i, \hat{\alpha}_i) = f_\theta(\mathbf{y}_{i1}, \mathbf{x}_i), \]

with \(f_\theta\) trained jointly with the dynamical model. This separates the question of what the dynamics are from the question of where a given subject sits within them.

For sparse designs this is the most useful member of the family. It is fully estimable at \(T = 2\)–\(3\) because the learned component operates on the cross-section, where information is abundant, while the dynamical structure is pooled across subjects. It also yields a directly interpretable clinical quantity: a predicted progression rate or stage from a single visit.

Minimum design: \(T = 2\) with large \(N\).

9.3 H3. Universal differential equations

A neural term is added to a mechanistic vector field:

\[ \frac{d\mathbf{x}}{dt} = -\beta \, \mathbf{L}\mathbf{x} + g_\theta(\mathbf{x}, \mathbf{u}), \]

with \(g_\theta\) a neural network. Training is by fitting the population, not by fitting individual subjects. The framework is reviewed for neuroscience applications by El-Gazzar and Gerven (2025).

Two documented difficulties apply directly to sparse designs. First, parameter inference is complicated by non-convex optimization landscapes, and single-start optimization fails reliably; data sparsity and noise substantially degrade performance, and the interpretability of the mechanistic parameters is compromised when the neural term absorbs mechanistic signal (Philipps et al. 2025). Second, mitigating these problems requires explicit regularization of the neural component toward physiologically plausible behavior (Rooij et al. 2025).

At \(T = 2\)–\(3\) the expected failure mode is that \(g_\theta\) absorbs measurement noise and the fitted \(\beta\) becomes uninterpretable. If this arm is included, it requires a penalty driving \(g_\theta\) toward zero, and an ablation showing what the neural term contributes beyond the mechanistic skeleton alone.

Minimum design: \(T \geq 4\) for a meaningfully constrained neural term; usable at \(T = 3\) only with strong regularization and honest reporting of the ablation.

9.4 H4. Constrained supervised learning

A supervised learner is trained with penalties enforcing properties implied by the mechanism — monotonicity, positivity, conservation of a total, consistency between horizons. This retains the flexibility of Chapter 8 while removing predictions that are mechanistically impossible.

This is a low-cost addition to the supervised arm and is frequently sufficient to capture most of the benefit attributed to more elaborate hybrids. It should be evaluated before H3.

Minimum design: as for Chapter 8.

9.5 H5. Latent variable dynamical models

Latent ODEs, ODE-RNNs, and related architectures learn a vector field in a latent space with an encoder mapping observations to initial conditions. These handle irregular sampling natively, which is an advantage in this setting.

They are nonetheless parameter-rich and, at \(T \leq 3\), the encoder is trained on essentially cross-sectional information while the latent dynamics are constrained by a single interval per subject. They belong to the dense regime (Chapter 13) unless the latent dimension is held very low and the result is reported with explicit comparison to H2, which they generalize.