This paper proposes Lax pairs informed neural networks (LPNNs) for solving integrable systems with Lax pairs. LPNNs are designed to leverage the structure of Lax pairs to efficiently solve data-driven localized wave solutions and associated spectral problems. Two versions of LPNNs are introduced: LPNN-v1 and LPNN-v2. LPNN-v1 transforms the solving of nonlinear integrable systems into solving linear Lax pairs spectral problems, enabling efficient data-driven solutions and extraction of spectral parameters. LPNN-v2 incorporates the compatibility condition/zero curvature equation of Lax pairs, enhancing the ability to solve high-accuracy localized wave solutions and spectral problems. The methods are validated through numerical experiments on several important integrable systems, including the KdV equation, Camassa-Holm equation, Kadomtsev-Petviashvili equation, and high-dimensional KdV equation. The results demonstrate that LPNNs significantly outperform conventional PINNs in terms of training efficiency and accuracy, particularly for systems with complex solutions. The integration of Lax pairs into deep neural networks provides a novel approach for studying data-driven localized wave solutions and Lax pairs spectral problems in integrable systems.This paper proposes Lax pairs informed neural networks (LPNNs) for solving integrable systems with Lax pairs. LPNNs are designed to leverage the structure of Lax pairs to efficiently solve data-driven localized wave solutions and associated spectral problems. Two versions of LPNNs are introduced: LPNN-v1 and LPNN-v2. LPNN-v1 transforms the solving of nonlinear integrable systems into solving linear Lax pairs spectral problems, enabling efficient data-driven solutions and extraction of spectral parameters. LPNN-v2 incorporates the compatibility condition/zero curvature equation of Lax pairs, enhancing the ability to solve high-accuracy localized wave solutions and spectral problems. The methods are validated through numerical experiments on several important integrable systems, including the KdV equation, Camassa-Holm equation, Kadomtsev-Petviashvili equation, and high-dimensional KdV equation. The results demonstrate that LPNNs significantly outperform conventional PINNs in terms of training efficiency and accuracy, particularly for systems with complex solutions. The integration of Lax pairs into deep neural networks provides a novel approach for studying data-driven localized wave solutions and Lax pairs spectral problems in integrable systems.