Numericical solutions are not solutions based directly on the underlying laws - https://wiki.tfes.org/Numerical_Solutions
Tom, again, this is demonstrably utter nonsense. Stepping back from the particular problem at hand, and just speaking generally, numerical methods are a broad class of computational technique. To categorise all of them as not being based on their underlying laws is false. There are numerous problems in physics and engineering (for example) where system dynamics are governed by time-based differential equations that simply cannot be solved algebraically. This fact does not invalidate the principles in question. Moreover, even if they could be solved by simple algebra, it's also important to realise that in many cases, planetary motion included, the resulting output would still have some degree of error due to the modelling assumptions made (eg point masses versus spheres, neglecting some objects as being too small to be significant) and small errors in starting conditions (mass estimation, velocity, position data etc). So to characterise numerical solutions as 'wrong' and algebraic solutions as 'correct' is in itself a flawed logic. In aerodynamics, for example, algebraic solutions of gas flow past solid bodies are usually quite poor, as the simplifications required to reduce the equations to something solvable involve removing important real-world effects such as viscosity and turbulence.
Your repeated assertion that numerical methods aren't based on the underlying laws is frankly absurd. Whilst there may well be examples of this, it is certainly not a valid statement for the totality of numerical methods. Computational fluid dynamics algorithms, for example, are absolutely based on the behaviours of gases and the equations that govern them. Numerical methods for solving flow past complex shapes typically involve breaking the flow field down into discrete elements, akin to pixels, and solving each tiny element's input and output flows in turn, with the whole system advanced in small discrete time-steps. This does of course introduce error, but the smaller the elements and the smaller the time-steps, the less error there is, and the closer to the truth we get. I mention aerodynamics simply because it's a field I'm trained in, but you can find examples of this kind of solution used in all manner of science, ranging from particle physics through to astronomy.
I've very much enjoyed reading around the JPL ephemeris models - not something I'd encountered before. It's abundantly obvious from reading into them that they are absolutely based on the underlying physics. The simpler models were essentially time-stepped newtonian solutions for point masses, whereas the more recent models exploit greater computational power and include all kinds of sophistications to better approximate planet shapes and internal composition, correct for relativistic effects, include small bodies such as asteroids etc...it's impressive science.
Your assertion that NASA's inclusions of Saros cycles is somehow indicative of failings or sleight of hand in their modelling is equally absurd. Yes, solar and lunar eclipses adhere to the Saros cycles - that's well understood. But Saros models are not completely accurate in terms of accurately geo-locating eclipse shadows, and it is clear that the various ephemeris models are now considerably more accurate in terms of predictive power.
You have not shown that the systems are actually based on the underlying laws
Ok then, let's go.
Here's the description of DE 102, one of the early and simpler models, taken from
https://link.springer.com/content/pdf/10.1007%2F978-94-009-7214-8_6.pdf:
A. Initial Conditions
The starting epoch of the integration was June 28, 1969 (JD 2440400.5), the ephemeris being integrated both forward and backward from this date. The initial conditions were the best available at that time and represented a least squares adjustment to a variety of observational data types. These included: 1) Lunar-laser ranging; 2) Mariner 9 and Viking Orbiter spacecraft ranging; 3) radar-ranging to Mercury, Venus, and Mars; and 4) Meridian circle optical data. These are described in detail in the paper cited above.
B. Equations of Motion
The equations of motion used in the integration included: 1) the n- body forces of the sun, moon, and the nine major planets; 2) the lunar librations; 3) isotropic, PPN-relativistic formulation; and 4) the perturbations from five asteroids. Though a number of the inherent constants have subsequently been modified, it is of importance to mention that the form of the equations of motion in DEl02/LE5l has not been changed in any o·f the more recent ephemerides produced at JPL.
Presumably, as a flat-earther, the lunar laser ranging and radar ranging of various planets is problematic for you? Likewise Mariner 9 and Voyager 9 data is equally conspiratorial?
I'm still waiting for that evidence, by the way. Maybe you could provide it in the same post that includes your eclipse predictive prognostication.