Provable Multi-Region Affinity Enforcement and Constraint Satisfaction for Scientific Machine Learning

Abstract: Neural networks have shown strong promise in scientific machine learning, but strictly enforcing boundary, interface, and safety constraints remains difficult. Soft penalties require careful tuning and do not guarantee exact satisfaction, while existing hard-constraint methods are typically specialized to particular equations or geometries. We introduce mPOLICE, a general framework that guarantees exact constraint satisfaction across multiple disjoint regions by exploiting the piecewise-linear structure of standard neural networks. By strategically configuring the network's internal activations, mPOLICE ensures that the learned function becomes exactly affine (linear) throughout each user-specified constrained zone. Once the network is locally affine, complex physical and safety constraints reduce to simple linear equations evaluated only at the corners (vertices) of each zone. Crucially, our approach handles many disjoint regions independently--a major limitation of existing single-region approaches. The method adds no inference-time overhead, requires no architectural changes, and integrates readily with standard training pipelines. We validate mPOLICE on operator learning, PDE boundary-condition enforcement, safety-critical control, and implicit 3D shape modeling.

Submission Type: Regular submission (no more than 12 pages of main content)

Assigned Action Editor: ~Bernhard_C_Geiger1

Submission Number: 8148