An Extreme Learning Machine-Based Method for Solving Linear–Quadratic Nonzero-Sum Differential Games

Multi-agent interaction in linear–quadratic (LQ) differential games gives rise to open-loop Nash equilibria that rarely admit closed-form expressions, motivating the development of reliable numerical solvers. Classical approaches such as shooting and spectral collocation are sensitive to the initial guess on the unknown boundary values and accumulate discretisation error over long horizons, while deep-learning alternatives require iterative gradient-based training with architecture- and convergence-specific overhead. To overcome these limitations, we recast the LQ nonzero-sum game as a linear two-point boundary value problem (TPBVP) via the Pontryagin maximum principle (PMP) and solve it with a single-layer feedforward neural network (SLFN) in which hidden-layer parameters are sampled once and fixed. The state and all player-specific costates are parameterised by random hidden features on a uniform time grid, the boundary conditions are appended as dedicated rows of the linear collocation system, and the output weights follow from a single Moore–Penrose pseudoinverse, entirely bypassing gradient-based iteration. For the scalar LQ optimal-control TPBVP, a residual-to-solution stability theorem converts the continuous equation and boundary residuals into uniform state, costate, control, and cost error bounds. Validation across two-player low- and high-dimensional benchmarks, a heterogeneous three-player game, and paired seed sweeps confirms high accuracy against analytical and matrix-exponential references, while revealing that no single activation function dominates across all problem types: tanh is most accurate in one-dimensional settings, and Gaussian RBF leads in multidimensional cases.