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Neural Networks (NNs) can be used to solve Ordinary and Partial DifferentialEquations (ODEs and PDEs) by redefining the question as an optimizationproblem. The objective function to be optimized is the sum of the squares ofthe PDE to be solved and of the initial/boundary conditions. A feed forward NNis trained to minimise this loss function evaluated on a set of collocationpoints sampled from the domain where the problem is defined. A compact andsmooth solution, that only depends on the weights of the trained NN, is thenobtained. This approach is often referred to as PINN, from Physics InformedNeural Network~\cite{raissi2017physics_1, raissi2017physics_2}. Despite thesuccess of the PINN approach in solving various classes of PDEs, animplementation of this idea that is capable of solving a large class of ODEsand PDEs with good accuracy and without the need to finely tune thehyperparameters of the network, is not available yet. In this paper, weintroduce a new implementation of this concept - called dNNsolve - that makesuse of dual Neural Networks to solve ODEs/PDEs. These include: i) sine andsigmoidal activation functions, that provide a more efficient basis to captureboth secular and periodic patterns in the solutions; ii) a newly designedarchitecture, that makes it easy for the the NN to approximate the solutionusing the basis functions mentioned above. We show that dNNsolve is capable ofsolving a broad range of ODEs/PDEs in 1, 2 and 3 spacetime dimensions, withoutthe need of hyperparameter fine-tuning.

dNNsolve: an efficient NN-based PDE solver