Locally adaptive Physics-Informed Neural Networks for nonlinear partial differential equations

Physics-Informed Neural Networks (PINNs) have emerged as a powerful paradigm for solving forward and inverse problems governed by partial differential equations (PDEs). However, standard PINNs often struggle with convergence and accuracy, especially when solving non-linear PDEs. This difficulty is largely attributed to the static nature of traditional activation functions, which cannot dynamically adapt to the evolving topology of the loss landscape during training. To address these limitations, this work investigates the integration of adaptive activation functions within the PINNs framework using both locally adaptive (neuron-wise and layer-wise), both with and without recovery term, to predict the solutions of non-linear PDEs. That is, we show both locally layer-wise (L-LAAF) and locally neuron-wise (N-LAAF) instantiation of adaptation of the activation functions with and without a recovery term in the loss function. We evaluate the proposed adaptive PINNs framework across a series of 4 benchmark problems. We observed improvements over the baseline solutions (obtained using standard PINNs) ranging from a minimum of 0.19% to a maximum of 92.84%, depending on the specific adaptive variant of PINNs employed.