Operator-Valued Neural Networks: Compositional Operator Learning for Physics and AI

We introduce Operator-Valued Neural Networks (OVNNs), a neural architecture where inputs, outputs, and layer activations are operators rather than numerical scalars or vectors. OVNNs can represent and learn complex operator transformations, preserving algebraic structures such as Hermiticity, unitarity, and commutation relations. This framework naturally encodes compositional rules, emergent dynamics, and structured interactions, with applications in physics-informed machine learning, multi-agent systems, and symbolic reasoning. We demonstrate the OVNN concept via a small symbolic example of a 3-spin lattice, highlighting operator composition, nonlinear functional layers, and expectation value outputs, and discuss its potential for increasing AI adaptability and learning efficiency.