Physics-informed quadratic residual neural network for solving forward and inverse problems of the two-dimensional time fractional convection–diffusion equation

Abstract This paper presents the physics-informed quadratic residual neural networks (PiQresNN) method for solving both the forward and inverse problems of the two-dimensional time-fractional convection–diffusion equation (TFCDE) in the Caputo sense. Since the Caputo fractional derivative does not satisfy the chain rule, it cannot directly apply automatic differentiation to compute derivatives as in the case of integer-order differential equations with the physics-informed neural networks (PINN) method. Based on this, the L 2 − 1 σ method with an approximation order of O ( τ 3 − α ) is embedded into PiQresNN to approximate the Caputo derivative, and further solve the two-dimensional TFCDE. Unlike traditional fully connected neural networks, the quadratic residual(Qres) neural network not only uses nonlinearity in the activation function but also introduces nonlinearity in the affine transformations inside the neurons (i.e., Qres neurons), which helps to better capture the nonlinear features of the equation. Moreover, by introducing an attention mechanism into the Qres neural network and incorporating two Transformer neural networks whose sizes match those of the hidden layers, the proposed PiQresNN strengthens the connections between the input and each hidden layer, thereby further enhancing its expressive capability. Based on three examples addressing both forward and inverse problems of TFCDE, the proposed method has been validated for its effectiveness. The results demonstrate the accuracy of the method, with prediction precision generally improving with an increase in hidden layers and neurons, and PiQresNN providing accurate solutions as evidenced by graphical representations of parameter variations and loss function dynamics.