Fig. 3. Solving a parametric diffusion-reaction system.

(Top) Exact solution versus the prediction of a trained physics-informed DeepONet for a representative example in the test dataset. (Bottom) Mean and SD of the relative L² prediction error of a trained DeepONet (with paired input-output training data) and a physics-informed DeepONet (without paired input-output training data) over 1000 examples in the test dataset. The mean and SD of the relative L² prediction are ∼1.92 ± 1.12% (DeepONet) and ∼0.45 ± 0.16% (physics-informed DeepONet), respectively. The physics-informed DeepONet yields ∼80% improvement in prediction accuracy with 100% reduction in the dataset size required for training. Tanh, hyperbolic tangent; ReLU, rectified linear unit.