The paper deals with fault isolation in nonlinear analogue circuits with tolerance under an insufficient number of independent voltage measurements. The L1-norm optimisation problem for different scenarios of nonlinear fault diagnosis is formulated with a new fast method being presented. How to solve the L1-norm optimisation problem is discussed and a new neural network-based approach for solving the nonlinear constrained L1-norm optimisation problem is proposed and utilised in locating the most likely faulty elements in nonlinear circuits. The validity of the proposed method is verified and simulation examples are presented.