Neural network generated intravoxel incoherent motion (IVIM) parameter maps are visually more detailed, less noisy and have a higher test-retest repeatability than conventional least-squares (LSQ) generated images. Recent literature suggests that tri-exponential models may fit liver IVIM data more accurately than conventional bi-exponential models, warranting investigation into the use of a neural network (NN) to estimate tri-exponential IVIM parameters. Here, we developed a tri-exponential IVIM unsupervised physics-informed deep neural network (IVIM3-NET), assessed its performance in non-alcoholic fatty liver disease (NAFLD) and compared outcomes with bi-exponential LSQ and NN fits and tri-exponential LSQ fits. Scanning was performed using a 3.0T free-breathing multi-slice diffusion-weighted single-shot echo-planar imaging sequence with 18 b-values. Images were analysed for visual quality, comparing the bi- and tri-exponential IVIM models for LSQ fits and NN fits using parameter-map signal-to-noise ratios (SNR) and adjusted R2. IVIM parameters were compared to histological fibrosis, disease activity and steatosis grades. Parameter map quality improved with bi- and tri-exponential NN approaches, with average parameter-map SNR increasing from 3.38 to 5.59 and 2.45 to 4.01 for bi- and tri-exponential LSQ and NN models respectively. In 33 out of 36 patients, the tri-exponential model exhibited higher adjusted R2 values than the bi-exponential model. Correlating IVIM data to liver histology showed that the bi- and tri-exponential NN outperformed both LSQ models for the majority of IVIM parameters (10 out of 15 significant correlations). Overall, our results support the use of a tri-exponential IVIM model in NAFLD. We show that the IVIM3-NET can be used to improve image quality compared to a tri-exponential LSQ fit and provides promising correlations with histopathology.