Scattering networks are deep convolutional architectures that use predefined wavelets for feature extraction and representation. They are mathematically well-understood, and have proven effective for classification tasks in limited training data scenarios, where traditional deep learning methods struggle. However, the opposite holds in larger data regimes, resulting in a performance gap between well-understood learning architectures and non-transparent yet highly effective paradigms. Our work addresses this gap on the domain of graphs by adapting the choice of diffusion operator that constructs the scattering network to the data, allowing better task-wise geometric representation. The resulting architecture preserves stability guarantees with respect to input perturbations. Continuous diffusion is applied in the learning process for more refined weight updates. Numerical experiments on benchmark datasets show that our approach consistently outperforms traditional graph scattering with predefined wavelets, expanding the scenarios where interpretable scattering architectures are competitive or superior to deep learning methods, and further reducing their aforementioned performance disparity.