Input Specific Neural Networks

I NPUT S PECIFIC N EURAL N ETWORKS A P REPRINT Asghar Jadoon The University of Texas at Austin Austin TX, USA D. Thomas Seidl Sandia National Laboratories Albuquerque NM, USA Reese E. Jones Sandia National Laboratories Livermore CA, USA Jan Fuhg The University of Texas at Austin Austin TX, USA February 2025 Abstract Neural networks have emerged as powerful tools for mapping between inputs and outputs. However, their black-box nature limits the ability to encode or impose specific structural relationships between inputs and outputs. While various studies have introduced architectures that ensure the network's output adheres to a particular form in relation to certain inputs, the majority of these approaches impose constraints on only a single set of inputs, leaving others unconstrained. This paper introduces a novel neural network architecture, termed the Input Specific Neural Network (ISNN), which extends this concept by allowing scalar-valued outputs to be subject to multiple constraints. Specifically, the ISNN can enforce convexity in some inputs, non-decreasing monotonicity combined with convexity with respect to others, and simple non-decreasing monotonicity or arbitrary relationships with additional inputs. To the best of our knowledge, this is the first work that proposes a framework that simultaneously comprehensively imposes all these constraints. The paper presents two distinct ISNN architectures, along with equations for the first and second derivatives of the output with respect to the inputs. These networks are broadly applicable. In this work, we restrict their usage to solving problems in computational mechanics. In particular, we show how they can be effectively applied to fitting data-driven constitutive models. We remark, that due to their increased ability to implicitly model constraints, we can show that ISNNs require fewer inputs than existing input convex neural networks when modeling polyconvex hyperelastic functions. We then embed our trained data-driven constitutive laws into a finite element solver where significant time savings can be achieved by using explicit manual differentiation using the derived equations as opposed to automatic differentiation. Manual differentiation also enables seamless employment of trained ISNNs in commercial solvers where automatic differentiation may not be possible. We also show how ISNNs can be used to learn structural relationships between inputs and outputs via a binary gating mechanism.