The solver is initialized with f , also called the free function. This initialization is often the first time point of the dynamics. To solve the IE and find the solution y , an iterative procedure is carried out in which at each solver step k , the integral of G ( y k , x , t ) is computed and used as the solution y k +1 in the next step. Integration is done either with Monte Carlo (via torchquad) or with self-attention, representing NIE and ANIE, respectively. The solver integration steps are repeated until convergence of y k to the IE solution. This solution is then compared with the input data to compute a loss thatvia backpropagationis used to find that minimizes the error. The resulting integral operator represents the IE that models the data. Top right: an example of attention weights for calcium imaging dynamics is presented. Bottom right: an example of the dynamical embedding of the NavierStokes dataset coloured by velocity is shown.
