Bayesian Embedding of Physical Knowledge within Continuous-Time Latent Neural Models for Dynamic Systems

Neural differential equations provide a flexible framework for learning dynamical systems from data, but can struggle with limited, noisy, and sparse data. Physical relations can provide a valuable inductive bias in these settings, yet existing soft constraints provide no inference-time guarantees, while hard state-space projections can be brittle under noise or misspecified physics. In this paper, we propose a Bayesian framework that embeds physical knowledge as functions of a vector field within a learned latent space, enforced via differentiable weighted projections. Performing joint Bayesian inference over both the model and physics-based parameters propagates uncertainty across the full framework. We show that, at each instant, latent projection approaches the same correction to the observed dynamics as state-space projection in the vanishing-regularization limit, but leaves the null space of the decoder Jacobian unconstrained. Across six forced dynamical systems, latent projection improves accuracy and calibration over unconstrained, soft-constrained and state-space-projected models when the physics is correct, and degrades far less than state-space projection when the embedded physics is misspecified.