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 with Weak Orthogonality Constraints for Enhanced Function and PDE Approximation&subtitle=We present polynomial-augmented neural networks (PANNs)%2C a novel machine learning architecture that combines deep neural networks (DNNs) with a polynomial approximant. PANNs combine the strengths of DNNs (flexibility and efficiency in higher-dimensional approximation) with those of polynomial approximation (rapid convergence rates for smooth functions). To aid in both stable training and enhanced accuracy over a variety of problems%2C we present (1) a family of orthogonality constraints that impose mutual orthogonality between the polynomial and the DNN within a PANN%3B (2) a simple basis pruning approach to combat the curse of dimensionality introduced by the polynomial component%3B and (3) an adaptation of a polynomial preconditioning strategy to both DNNs and polynomials. We test the resulting architecture for its polynomial reproduction properties%2C ability to approximate both smooth functions and functions of limited smoothness%2C and as a method for the solution of partial differential equations (PDEs). Through these experiments%2C we demonstrate that PANNs offer superior approximation properties to DNNs for both regression and the numerical solution of PDEs%2C while also offering enhanced accuracy over both polynomial and DNN-based regression (each) when regressing functions with limited smoothness.?quality=80&w=800)
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