However, our modern understanding of linear and non-linear systems do not permit this under many situations. Any theory which is non-linear will admit multiple solutions. Which solution actually happens in the future cannot be calculated from the equations of that theory. Two examples are the non-linearity in the Navier-Stokes equations. (whether Navier-Stokes has unique solutions is an unresolved problem in mathematics). Electromagnetism is non-linear. More shocking : Newton's mechanics are non-linear , and so situations can be constructed where many possible outcomes could happen, all consistent with the equations.

One proponent of this problem is an american professor, John D Norton who chairs the Philosophy of Science department at Pittsburgh. More detail can be found here : http://philsci-archive.pitt.edu/2943/1/Norton.pdf

In summary, not even classical physics is deterministic.

Things are about to get doubly ironic.

Word on the street is that Quantum Mechanics is a linear theory. This means QM is the only known physical theory that both admits unique solutions and allows for actual determinism --- Laplacian style determinism.

A caveat disclaimer should be inserted here. This is Quantum Mechanics in its unitary evolution, without observer events and other wave-function-collapsing oddities. This the "raw form" of the Schroedinger Equation. Many-a-reader have likely heard about quantum vacuum fluctuations, and other random aspects of quantum mechanics , such as the "measurement problem". A regular on this forum even went as far as to conclude :

Nature has at its disposal, in every fermion, a perfect random value generator - far better than any possessed by a digital computer. Our brains may well make use of this, perhaps relating to free will and the unpredictability of individual behaviors.

This so-called "perfect random value generator" is not part of any known equation or theory in physics. If for example, you consider Many-worlds or even quantum Bayesianism, those are from a cluster of interpretations that demand that the wave function never actually collapses. In other threads, I described these as No-Collapse Interpretations. These sit opposite from a cluster of interpretations called Collapse Theories.

Proponents of MWI, such as Max Tegmark, have often advertised Laplacian determinism as being a selling-point for MWI. I will try to link some lectures where Tegmark makes these kinds of claims... time permitting.