the heat equation instantly smooths any initial conditions, so that after an arbitrarily small amount of time t>0, the function which obeys the heat equations is infinitely differentiable in its spacial inputs. however, continuous functions which are nowhere differentiable are dense in the space of all continuous function, so given information about a function which is arbitrarily precise, you cannot reasonably determine whether it is smooth or not.

so if there is any physical process in the universe governed by the heat equation (which I believe is implied by some use of Atiyah-Singer in physics), then you can't tell the difference between a universe that has been evolving according to the heat equation, or a universe which started t>0 seconds ago from a nowhere differentiable set of initial conditions (which necessarily cannot flow backwards in time)