In ongoing work with Ellingsen, Heinrich and Seth, we rediscover a KdV-type equation from simple video recordings of shallow-water solitons. Using two independent data-driven methods — weak-form sparse regression (WSINDy) and a Fourier multiplier approach — we extract the same governing equation, despite noise and lack of control in the experimental setup. This suggests that dynamics are robustly encoded even in rough experimental data. In this talk I will further reflect on how data-driven techniques can support analysis of PDEs in settings where the underlying dynamics are unknown. I will also describe a separate line of work on numerically exact analysis using interval arithmetic, which rigorously proves existence of waves appearing at very subtle parameter relations. Together, these approaches point toward a future where data, computation and analysis play new and complementary roles in both modelling and mathematical deduction. More generally I will touch upon the role research mathematics can play in a world where AI has moved beyond today’s capabilities.