Shape-Preserving Networks: Encoding Structural Knowledge through Derivative Constraints and Nested Integration

Miguel Suau, Alec Edwards, Alex Durkin, Ian Howell, Giuseppe Pinto, Marc Artola, Daniel Jarne Ornia, Sayam Kumar

Keywords: shape-preserving networks, out-of-distribution generalization, physics, structure, derivative constraints, nested integration

Abstract: We present Shape-Preserving Networks (SPNs), a neural network architecture that guarantees monotonicity, curvature, and interaction structure by construction through nested integration. Unlike Physics-Informed Neural Networks (PINNs), where physics is enforced by embedding the governing physical laws as penalties directly into the loss function during training and hence provide no guarantees, or through explicit inequality constraints, SPNs encode derivative constraints directly into the forward pass, ensuring global satisfaction across the entire domain. The architecture decomposes functions into univariate and pairwise terms, constructing each from its derivatives with prescribed signs. We evaluate SPNs on synthetic benchmarks, a real-world chiller power prediction task, and a pendulum world model for model-based control. Results demonstrate physically plausible out-of-distribution extrapolation, improved robustness to outliers, and greater stability across training runs compared to standard networks, while the additive decomposition provides interpretable components that align with known physics.

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Submission Number: 7