2026 - M-Layer ML for Science Learning

Thomas, Jyrki, (Lab)


The Topic

Newton said gravity is a force; Einstein said gravity is the curvature of spacetime
Does Newton's Gal(3) invariant still hold at the galactic scale?
  • Negative samples cannot be arbitrarily noiseed: The position cannot be changed arbitrarily while maintaining the original acceleration, because arbitrarily adding noise would break the Gal(3) invariant, and the classifier would learn not the physical symmetry breaking, but noise
We only show SO(2) and SO(1,1)，SO(3), SE(3), SO(3,1) for 3D galaxies and general relativity is future work


Foundation of ML

Mercer's theorem guarantees that we do not need to explicitly write out or calculate the high-dimensional feature phi(x). We can directly calculate the scalar function k(x, x'), which is mathematically 100% equivalent to building a linear model in a certain feature space






End Goals

  • Pretty visuals
  • M-Layer for verifying symmetries induced by scientific laws in high-dimensional observation space.
  • What might be missed in current Theoretical Laws?

Experiment test

  • Can the model determine whether X=(q,v,a) is a valid physical state?


Other Topics

  • đź“Ť Unifying fermions and bosons
  • new gravitational-wave
  • Incorporating Maxwell’s Equations that describe the electromagnetic field into the Dirac Equation has been a long-term goal in physics to unify quantum electrodynamics (QED) and general relativity


Path to M²-Layer

Core Concept: M²-Layer = exp(Mb) * exp(Ma)

  1. Observe and define the physical state. In the current experiment, each observation is a Lorentz-state pair $x = (v, E)$. The physical manifold is defined by: \(E^2(1 - v^2) = 1\) with $y = 0$ denoting physical samples and $y = 1$ denoting non-physical samples. The model therefore acts as a physical-law verifier rather than a full dynamical simulator.

  2. Learn the original M-Layer generator. The current M-Layer constructs a state-dependent matrix: \(M_\theta(x) = \sum_i x_i T_i\) and evaluates a matrix-exponential score: \(r_\theta(x) = \mathrm{Tr}(\exp(M_\theta(x))S)\) The physical law is represented as an implicit zero set $r_\theta(x) = 0$. Thus, the current model is an algebraic physical-law verifier, not yet a one-step ODE/PDE flow map.

  3. Upgrade to a composed M²-Layer. The next architecture should not square the matrix literally as $\exp(M^2)$. Instead, it should compose two learned M-Layer flows: \(g_\theta(x) = \exp(M_b(x))\exp(M_a(x))\) By the Baker–Campbell–Hausdorff expansion: \(\exp(M_b)\exp(M_a) = \exp\left(M_a + M_b + \frac{1}{2}[M_b, M_a] + \dots\right)\) where: \([M_b, M_a] = M_b M_a - M_a M_b\) The commutator term introduces curvature and non-commutative interactions that a single exponential generator cannot express. This is the natural next step suggested by the hard-negative failure in the current ablation.


Three Key Metrics

  • ROC-AUC
    • Does the learned score separate physical from unphysical samples across thresholds?
  • PoincarĂ© section consistency
    • Recover the same phase-space section geometry as the true dynamical system.
  • Zero-shot / Systematic Generalization / (Out-of-Distribution (OOD) Generalization Ability)



Lorentz data

  • 1D
gamma / energy
   ^
   |                              /
   |                            /
   |                         /
   |                      /
   |                  /
   |             __/
   |        __/
   |______/________________________> beta = v/c
        0        0.5       0.9   1


  • 2D
beta_y
  ^
  |        high gamma near boundary
  |       ***********************
  |     ***                     ***
  |    **       low gamma         **
  |    **        near center      **
  |     ***                     ***
  |       ***********************
  +-------------------------------> beta_x


When Choose Which Structure

Space Type Example Original M-Layer M²-Layer
Algebraic constraint space ((v,E)), ((p,E)) Enough Optional
Simple state space ((q,v,a)) one law Often enough Optional
Phase space ((q,p)) with multiple invariants May fail Useful
Function space (u(t,x)) PDE field Limited Useful
Operator / Hilbert space quantum (H_a,H_b) Limited Very useful
Gauge / curvature space field connection, commutator curvature Not enough Needed


What Else Can be Verified?

Frontier Problem Core Open Question Why AI / M-Layer Direction
Quantum Gravity How can general relativity and quantum mechanics be unified? It asks what spacetime is at the Planck scale. Learn whether observed dynamics lie on hidden constraint manifolds consistent with classical, semiclassical, or beyond-GR regimes.
Dark Matter Is dark matter a particle, a field, primordial black holes, or modified gravity? Most matter in the universe is gravitationally visible but physically unidentified. Use high-dimensional classifiers to distinguish Newtonian gravity, dark matter halos, and modified-gravity trajectories from lensing and galaxy dynamics.
Dark Energy Is cosmic acceleration caused by a cosmological constant or evolving dark energy? Recent DESI results strengthen hints that dark energy may evolve over time. (DESI) Learn deviations from ΛCDM as geometric constraints in cosmological trajectory space.
Black Hole Information Problem Does black hole evaporation preserve quantum information? It tests the consistency of quantum mechanics, thermodynamics, and gravity. Use AI as a verifier for whether simulated black-hole dynamics obey conservation, entropy, and causal constraints.
Testing General Relativity Does Einstein gravity remain exact in strong-field regimes? Gravitational waves allow direct tests near merging black holes. Current LIGO-Virgo-KAGRA analyses remain consistent with GR. (aei.mpg.de) Train physical-law verifiers on waveform manifolds to detect subtle beyond-GR deviations.
Gravitational-Wave Discovery Can we extract weak, noisy, rare signals from detector data? It opens a new observational channel for black holes, neutron stars, and early-universe physics. AI is already used for denoising, waveform modeling, and parameter estimation in gravitational-wave pipelines. (Hep Journals)
Cosmic Structure Formation How did galaxies, halos, filaments, and voids emerge from early fluctuations? It connects gravity, dark matter, baryons, and cosmology across billions of years. Learn compact latent laws from simulations and surveys; explainable ML has already found interpretable structure in dark-matter halo profiles. (mpa-garching.mpg.de)
Weak Lensing and Hidden Mass Can we reconstruct invisible matter from distorted galaxy images? Lensing is one of the cleanest probes of dark matter and dark energy. Use AI-assisted simulation-based inference and constraint learning to map hidden mass fields from sparse observations. (Imperial College London)
Modified Gravity Are cosmic anomalies caused by unseen matter or by a failure of GR at large scales? It challenges the foundation of modern cosmology. Build contrastive datasets: GR-valid trajectories vs modified-gravity trajectories, then learn the separating physical manifold.
From Galaxies to Cells Can one learning principle discover lawful dynamics across scales? It would unify scientific ML beyond domain-specific prediction. M-Layer can be framed as a high-dimensional physical constraint learner: not predicting one trajectory, but verifying whether a system belongs to the lawful manifold.



Physical Spaces

Layer Physical / Mathematical Theory Proposed by / Historical Origin Year / Period Representative Space Representative Dimension Core State Variable Core Law / Object Relevance to GOES / NOAA Energetic-Particle Data Meaning for M-Layer / Stage III
1 Euclidean physical space Euclid c. 300 BCE Classical geometric space 3D Position $(\mathbf{x}\in\mathbb{R}^3)$ Distance, angle, straight line The basic spatial background for satellite position, Earth-centered coordinates, and magnetospheric geometry The lowest-level space in which GOES occupies a point on a geostationary orbit
2 Newtonian absolute space and time Isaac Newton 1687 Absolute space plus absolute time (3+1)D $(\mathbf{x},t)$ $\mathbf{F}=m\mathbf{a}$, inverse-square gravity Provides the classical orbital description of GOES around Earth Useful for satellite ephemeris, but not sufficient for plasma-field reconstruction
3 Keplerian orbital geometry Johannes Kepler 1609–1619 Conic-section orbital space 2D orbital plane embedded in 3D Orbital elements $(a,e,i,\Omega,\omega,M)$ Elliptic motion, area law, harmonic law Describes idealized satellite/planetary motion before perturbations Gives a controlled synthetic benchmark, but not the real MHD field
4 Lagrangian mechanics Joseph-Louis Lagrange 1788 Configuration manifold and tangent bundle $n$-D configuration; $2n$-D tangent state $(q,\dot{q})$ Euler–Lagrange equations Useful for formulating particles moving under electromagnetic fields Connects observed particle motion to hidden variational structure
5 Hamiltonian mechanics William Rowan Hamilton 1833–1835 Phase space $2n$-D for $n$ degrees of freedom $(q,p)$ Hamiltonian flow $(\dot{q}=\partial H/\partial p, \dot{p}=-\partial H/\partial q)$ Charged-particle motion in magnetic fields is naturally Hamiltonian or nearly Hamiltonian M-Layer can be interpreted as learning a generator of flow rather than a static classifier
6 Symplectic geometry Hamilton, Jacobi, Poincaré; modern formalization by Hermann Weyl and others 19th–20th century Symplectic manifold          



ODE

Person / period Year Problem they cared about Equation style Essence
Newton / Leibniz 1660s–1670s Motion, curves, and rates of change $\frac{dy}{dx}=f(x,y)$ Differential equations emerge from calculus
Newton 1687 Planetary motion and mechanics $m\ddot{q}=F(q)$ Physical motion as time evolution
Jacob Bernoulli / Leibniz 1695–1696 Solvable nonlinear first-order equations $y’ + P(x)y = Q(x)y^n$ Early named ODE family
Euler 1700s, especially 1730s–1760s Mechanics, astronomy, fluids, and variational problems Systematic ODE methods ODEs become a general mathematical tool
Cauchy Early 1800s Existence, uniqueness, and initial-value problems $y(t_0)=y_0$ Foundation of modern rigorous ODE theory


PDE

Person / period Year Problem they cared about Equation style Essence
d’Alembert 1747 Vibrating strings $u_{tt}=c^2u_{xx}$ Wave propagation
Euler / Daniel Bernoulli 1740s–1750s Strings, fluids, and continuum mechanics Early PDE models Motion of continuous media
Laplace Late 1700s Gravitational and electrostatic potential $\Delta \phi=0$ Equilibrium fields
Fourier 1807 / 1822 Heat conduction $u_t=\kappa u_{xx}$ Diffusion of a temperature field
Poisson 1810s Potential fields with sources $\Delta \phi=f$ Fields generated by matter or sources
Navier / Stokes 1822 / 1845 Viscous fluid flow Navier–Stokes equations Evolution of fluid velocity fields
Maxwell 1861–1865 Electricity and magnetism Coupled PDEs for $\mathbf{E}$ and $\mathbf{B}$ fields Electromagnetic field theory
Einstein 1915 Gravity as spacetime geometry Einstein field equations Dynamics of the spacetime metric field



Simulation

  1. đź“Ť NASA GMAT
  2. Raindrops
trajectory generation
mission scenario
orbit propagation
lunar / deep-space transfer
navigation analysis

GMAT / Basilisk generates trajectory
→ export trajectory points
→ CesiumJS / Three.js visualizes it beautifully



Visualization

  1. NASA Eyes / Eyes on the Solar System
  2. Houdini
3D scene
+ time controller
+ camera fly-through
+ mission trajectory
+ object labels
+ information panel



Web Interaction

  1. NASA/mission-viz
    • link
    • CesiumJS / Three.js




























References



Bio process

  • 1 2012, Analysis, Synthesis, and Design of Chemical Processes.



Lie Group, ODE, and Chaos

  • 1 Georgi, Howard. Lie Algebras in Particle Physics. 2nd ed. Boca Raton: CRC Press, 2018.
  • đź“Ť 2 Hairer, Ernst, and Gerhard Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. 2nd rev. ed. Berlin: Springer, 2010.
  • 3 Sussman, Gerald Jay, and Jack Wisdom. Structure and Interpretation of Classical Mechanics. 2nd ed. Cambridge, MA: MIT Press, 2015.
  • đź“Ť 4 Strogatz, Steven H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. 2nd ed. Boulder: Westview Press, 2018. (Nonlinear Dynamics / Benchmark ODE)
  • 5 Hall, Brian C. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. 2nd ed. Cham: Springer, 2015.



References 2


Other Topics


References