Gradient-informed neural networks: Embedding prior beliefs for learning in low-data scenarios

We propose Gradient-Informed Neural Networks (GradINN s), a methodology that can be used to efficiently approximate a wide range of functions in low-data regimes, when only general prior beliefs are available, a condition that is often encountered in complex engineering problems. GradINN s incorporate prior beliefs about the first-order derivatives of the target function to constrain the behavior of its gradient, thus implicitly shaping it, without requiring explicit access to the target function's derivatives. This is achieved by using two Neural Networks: one modeling the target function and a second, auxiliary network expressing the prior beliefs about the first-order derivatives (e.g., smoothness, oscillations, etc.). A customized loss function enables the training of the first network while enforcing gradient constraints derived from the auxiliary network; at the same time, it allows these constraints to be relaxed in accordance with the training data. Numerical experiments demonstrate the advantages of GradINN s, particularly in low-data regimes, with results showing strong performance compared to standard Neural Networks across the tested scenarios, including synthetic benchmark functions and real-world engineering tasks.