Minimal Filling Architectures of Polynomial Neural Networks: Counterexamples, Frontier Search, and Defects

We provide a counterexample to the minimal unimodal conjecture for polynomial neural networks (PNNs) with power activation functions. Fixing the input and output widths, the conjecture states that any minimal filling architecture has unimodal widths for the hidden layers. We found a counterexample via a frontier search and certified it using recursive dimension bounds and symbolic computation. Notably, several subarchitectures of this example exhibit large defect, in contrast with the predominantly small-defect behavior observed in prior examples.

Rule-based autocorrection of Piping and Instrumentation Diagrams (P&IDs) on graphs

A piping and instrumentation diagram (P&ID) is a central reference document in chemical process engineering. Currently, chemical engineers manually review P&IDs through visual inspection to find and rectify errors. However, engineering projects can involve hundreds to thousands of P&ID pages, creating a significant revision workload. This study proposes a rule-based method to support engineers with error detection and correction in P&IDs. The method is based on a graph representation of P&IDs, enabling automated error detection and correction, i.e., autocorrection, through rule graphs. We use our pyDEXPI Python package to generate P&ID graphs from DEXPI-standard P&IDs. In this study, we developed 33 rules based on chemical engineering knowledge and heuristics, with five selected rules demonstrated as examples. A case study on an illustrative P&ID validates the reliability and effectiveness of the rule-based autocorrection method in revising P&IDs.