Physics Informed Neural Networks, Discrete Versus Continuous Functions, Fractal Dimensions, and Nature

There was a recent nanoHUB discussion of Physics Informed Neural Networks (PINNs) and phase difference calculations utilizing differential relations described by interacting nodes in a connectionist paradigm of discrete error correction, generating a continuum of solution sets for a description of a physical system. The question at the heart of the discussion was whether a digital and discrete logic circuit - be it transistor array, quantized qubit Hamiltonian, or PINN - can accurately describe the continuous nature of a physical reality. By overlapping discrete values, one can approximate and estimate a continuous functional space, similar to the theorem of calculus, yet the question gets to something deeper. Is all space quantized? Quantum mechanics and general relativity are critical to understand physics in the modern era. Quantum posits a wave equation, probabilistic eigenmode collapse and what is most likely pseudo-randomness. General relativity declares dimensions are parts of a spacetime. What is interesting is thinking of a new type of space as self-similar where the Green's functions, fields, points, and waves of spacetime are all correlated to an invariant of scale. What happens in the small scale is transmitted or reflected into the larger scale. Nature is this seemingly self-similar architecture where the patterns that emerge (and that can be observed) are related to a completeness relation of Gödel. Basically, the complex adaptive system measures with photons and field interaction, phenomena, such that everything is connected topologically.

Continuous connections and affine connections have implications in PINNs such that new A.I. descriptions of nature will involve the approximations of physical space where variables are smoothed over in a Taylor expansion fitting. What this means is that nature is measured as discrete and described as discrete, with only emergent properties being continuous. Quantum seems to win. Thus, Yang-Mills theory suggests that the nature of the infinitesimal is related to the integer representation of matrix states all the way down. At the bottom is a photon of light that manifests as energy, which is spatial curvature which is mass. The self similar nature of everything means that the CMB field (2.7K) from the BIg Bang in the hyper-dimensional picture is correlated to the inflationary universe of today. The self-similarity and scale invariance are connected to the time function. There is an additional multiplicity of time, what is called nT time, which suggests multiple bifurcating timelines. The application is in a grand reworking of least action geodesics and a better description of the Standard Model which removes superfluous particles. In all likelihood, there are fewer particles, not more, as all things are Yang-Mills states operated upon by Green's function tensor products of amplitude function that expand in space like fractals.

Measurement of this in an experiment requires sensitive phased array radar or phased array quantum sensors that can absorb information (photons), their phase, their location, their time, and trace the paths through some physical system. If one particle is difficult to trace, maybe the bulk ensemble can be traced and the extrapolation (such as the one done at CERN) might elucidate a Pythagorean and Euclidean like topological solution to the fractal shape of spacetime.