This paper presents a neural network-based digital calibration algorithm for high-speed and time-interleaved (TI) ADCs. In contrast with prior methods, the proposed work features joint amplitude-dependent and phase-dependent nonlinear distortion correction without prior-knowledge of ADC architecture feature. A dynamic calibration is first used to compensate for phase-dependent distortion. Two training optimizations, including a sub-range-sample-based batch schemes and a recursive foreground co-calibration flow are proposed to reduce the error and overfitting and further save hardware resources. A practical calibration engine is also investigated for interleaved ADCs with distributed weight and shared weight methods. To demonstrate the effectiveness of the method, the calibration engine is verified by two fabricated ADC prototypes, a 5 GS/s 16-way interleaved ADC and a 625 MS/s interleaving-SAR assisted pipeline ADC. Measurement results show that SFDR is improved between 16.9dB and 36.4dB before and after calibration for different frequency inputs. To trade-off between accuracy and power consumption, a quantized and pruned engine is implemented on both FPGA and 28nm CMOS technology. Experimental results show that the dedicated calibration on silicon consumes 8.64mW with 0.9V power supply at 333MHz clock rate. Measurement results show that the quantized hardware implementation has only 0.4-4 dB loss in SFDR.