2026 - Important Thesis - M-Layer on Scientific Law Verification, ODE to PDEs

Thomas, Jyrki, profs and Labs



End Goals

  • pretty visuals
  • M-Layer for discovering and verifying 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 xxx?

Key Index:

  • 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


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.


High-Dimensional Constraints Induced by Scientific Laws

Domain Scientific Law / Equation Formula Physical Meaning What M-Layer Can Verify or Learn        
Special Relativity Mass-energy equivalence $E = mc^2$ Mass and energy are equivalent; rest mass is a form of energy. Verify whether generated particle or event data preserve relativistic energy relations.        
Special Relativity Relativistic energy-momentum relation $E^2 = p^2c^2 + m^2c^4$ Energy, momentum, and mass are constrained by spacetime geometry. Check whether simulated high-energy particles lie on the relativistic mass shell.        
Special Relativity Lorentz factor $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$ Time dilation and length contraction depend on velocity. Detect impossible trajectories where $v > c$ or relativistic timing becomes inconsistent.        
Special Relativity Time dilation $\Delta t’ = \gamma \Delta t$ Moving clocks are measured differently by different observers. Verify relativistic time-series data, such as satellite timing or high-speed motion.        
Special Relativity Length contraction $L = \frac{L_0}{\gamma}$ Moving objects contract along the direction of motion. Detect violations in generated relativistic geometry.        
Special Relativity Four-velocity normalization $u^\mu u_\mu = -c^2$ Physical worldlines have a fixed spacetime norm. Learn the lawful manifold of admissible relativistic trajectories.        
General Relativity Einstein field equations $G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}$ Matter-energy determines spacetime curvature, and curvature determines motion. Verify whether a proposed metric $g_{\mu\nu}$ and stress-energy tensor $T_{\mu\nu}$ are mutually consistent.        
General Relativity Einstein tensor $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}$ Curvature is summarized by the Einstein tensor. Test whether learned geometric fields satisfy curvature consistency.        
General Relativity Vacuum Einstein equation $R_{\mu\nu} = 0$ Empty spacetime can still contain curvature, such as gravitational waves or black holes. Verify whether simulated vacuum spacetime solutions are physically admissible.        
General Relativity Geodesic equation $\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0$ Free-falling objects follow geodesics in curved spacetime. Verify whether trajectories are consistent with an inferred gravitational field.        
General Relativity Stress-energy conservation $\nabla_\mu T^{\mu\nu} = 0$ Energy and momentum are locally conserved in curved spacetime. Verify whether field simulations preserve local conservation laws.        
General Relativity Bianchi identity $\nabla_\mu G^{\mu\nu} = 0$ Geometric consistency condition underlying energy-momentum conservation. Test whether learned curvature fields obey internal geometric consistency.        
Cosmology Friedmann equation $H^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}$ Describes the expansion rate of the universe. Verify whether cosmological trajectories match GR-based expansion laws.        
Cosmology Acceleration equation $\frac{\ddot a}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}$ Determines whether cosmic expansion accelerates or decelerates. Detect whether observed expansion history implies dark energy, modified gravity, or inconsistent dynamics.        
Cosmology Critical density $\rho_c = \frac{3H^2}{8\pi G}$ Defines the density required for a spatially flat universe. Verify whether inferred cosmological parameters lie in physically consistent regions.        
Gravitational Lensing Weak deflection angle $\alpha \approx \frac{4GM}{c^2b}$ Massive objects bend the path of light. Verify whether lensing patterns are consistent with visible mass, dark matter, or modified gravity.        
Black Holes Schwarzschild radius $r_s = \frac{2GM}{c^2}$ Defines the event horizon radius of a non-rotating black hole. Verify whether generated black-hole parameters are physically admissible.        
Black Holes Schwarzschild metric $ds^2 = -\left(1-\frac{2GM}{rc^2}\right)c^2dt^2 + \left(1-\frac{2GM}{rc^2}\right)^{-1}dr^2 + r^2d\Omega^2$ Exact GR solution for a spherical, non-rotating mass. Test whether simulated or inferred spacetime geometry matches the Schwarzschild solution.        
Black Holes Kerr spin bound $a = \frac{J}{Mc}, \quad a \leq \frac{GM}{c^2}$ Rotating black holes have bounded angular momentum. Detect generated rotating black holes that violate physical spin constraints.        
Gravitational Waves Linearized Einstein equation $\Box \bar h_{\mu\nu} = -\frac{16\pi G}{c^4}T_{\mu\nu}$ Weak gravitational waves are perturbations of spacetime. Verify whether waveform data are consistent with GR wave propagation.        
Gravitational Waves Vacuum wave equation $\Box \bar h_{\mu\nu} = 0$ Gravitational waves propagate through vacuum at light speed. Detect non-GR waveform deviations or unphysical wave speeds.        
Gravitational Waves Quadrupole radiation principle $P \sim \frac{G}{5c^5}\left\langle \dddot Q_{ij}\dddot Q_{ij}\right\rangle$ Gravitational waves are mainly emitted by changing mass quadrupoles. Verify whether simulated binary systems radiate energy consistently.        
Electromagnetism / Field Theory Gauss law for electricity $\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$ Electric charge is the source of electric fields. Verify whether learned electric fields are consistent with charge distributions.        
Electromagnetism / Field Theory Gauss law for magnetism $\nabla \cdot \mathbf{B} = 0$ Magnetic monopoles are absent in classical Maxwell theory. Detect unphysical magnetic fields with nonzero divergence.        
Electromagnetism / Field Theory Faraday induction law $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ Time-varying magnetic fields induce electric fields. Verify whether temporal field evolution obeys electromagnetic induction.        
Electromagnetism / Field Theory Ampere-Maxwell law $\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t}$ Currents and time-varying electric fields generate magnetic fields. Learn whether electromagnetic field simulations preserve Maxwell dynamics.        
Electromagnetic Waves Wave equation in vacuum $\Box A^\mu = 0$ Light is a propagating electromagnetic field. Verify whether generated wave fields propagate causally at speed $c$.        
Gauge Field Theory Electromagnetic field tensor $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ Electromagnetic fields arise from a gauge potential. Learn gauge-consistent field representations instead of only fitting raw fields.        
Quantum Mechanics Schrodinger equation $i\hbar\frac{\partial}{\partial t}\psi = \hat H\psi$ Quantum states evolve under the Hamiltonian operator. Verify whether learned quantum-state trajectories obey unitary evolution.        
Quantum Mechanics Born rule $P(x) = \psi(x) ^2$ Measurement probabilities are given by wave-function amplitudes. Check whether generated quantum states produce valid probability distributions.    
Quantum Mechanics Canonical commutation relation $[\hat x, \hat p] = i\hbar$ Position and momentum cannot both be sharply determined. Verify whether learned latent operators preserve quantum structure.        
Quantum Field Theory Klein-Gordon equation $(\Box + m^2)\phi = 0$ Relativistic scalar fields satisfy a wave-like mass-shell equation. Verify whether generated scalar fields satisfy relativistic field constraints.        
Quantum Field Theory Dirac equation $(i\gamma^\mu\partial_\mu - m)\psi = 0$ Relativistic spin-1/2 particles are described by spinor fields. Learn physically admissible fermionic field dynamics.        
Quantum Field Theory Yang-Mills field equation $D_\mu F^{\mu\nu} = J^\nu$ Non-Abelian gauge fields generalize electromagnetism. Verify whether learned gauge fields obey local symmetry constraints.        
Standard Model Gauge symmetry structure $SU(3)_C \times SU(2)_L \times U(1)_Y$ The Standard Model is organized by local gauge symmetries. Learn whether high-dimensional particle interaction data respect symmetry-induced constraints.        
Higgs Mechanism Higgs potential $V(\phi) = \mu^2 \phi ^2 + \lambda \phi ^4$ Spontaneous symmetry breaking gives particles mass. Verify whether learned particle-field configurations lie near physically meaningful vacuum structures.
Statistical Physics Boltzmann distribution $p(x) = \frac{1}{Z}e^{-E(x)/(k_BT)}$ Equilibrium probabilities depend exponentially on energy. Check whether generated molecular or thermodynamic states obey equilibrium statistics.        
Brownian Motion Einstein diffusion relation $\langle x^2(t)\rangle = 2Dt$ Microscopic random motion produces macroscopic diffusion. Verify whether stochastic particle, cell, or molecular trajectories obey diffusion scaling.        
Statistical Physics Einstein mobility-diffusion relation $D = \mu k_BT$ Diffusion and mobility are linked by temperature. Test whether noisy dynamics respect thermodynamic constraints.        
Photoelectric Effect Photon energy $E = hf$ Light energy is quantized into photons. Verify whether photon-material interaction data obey quantum energy constraints.        
Photoelectric Effect Photoelectric equation $K_{\max} = hf - \phi$ Electron kinetic energy depends on light frequency and material work function. Verify whether simulated photoelectric data obey quantum threshold behavior.        
Atomic Radiation Detailed balance condition $N_1B_{12}\rho(\nu) = N_2B_{21}\rho(\nu) + N_2A_{21}$ Radiation and matter reach equilibrium through balanced transitions. Check whether learned atomic transition models preserve equilibrium physics.        







Toolkit

  • [npzviewer]

1. ODE




2.Hamiltonian Systems and Operators

  • Hamiltonian system
  • Hilbert space: Transition from state point to function space
  • Weierstrass Approximation Theorem
    • Universal Approximation Theorem, Bézier Curve
  • Lie operator, Unitary operator
  • Flow map




3. Topological Defect

  • The data manifold is extremely distorted, dimensionally reduced, or has broken connectivity in this region.




4. PDE / MHD, or Benchmark paper

learning hidden physical fields from sparse multimodal observations

multimodal observation
magnetic-field inference
streamer reconstruction
MHD-constrained dynamics
future disk prediction

ALMA / MHD simulation infers hidden protostellar dynamics
→ export magnetic fields, streamers, and gas-flow trajectories
→ CesiumJS / Three.js visualizes magnetogravitational accretion beautifully




5. Activation Functions and Field Theories











Simulation

  1. 📍 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 
3D scene
+ time controller
+ camera fly-through
+ mission trajectory
+ object labels
+ information panel



Web Interaction

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












Bio Process Dynamics: High-Dimensional Constraints Induced by Biochemical Process Laws

Domain Scientific Law / Equation Formula Physical Meaning What M-Layer Can Verify or Learn
Bioprocess Dynamics General process ODE manifold $\dot z = f(z,u;\theta)$ A bioprocess trajectory is governed by state-dependent kinetic and control laws. Verify whether a candidate point $(z,u,\dot z)$ lies on the learned dynamic process manifold rather than only matching observed trajectories.
Bioprocess Dynamics Bioprocess residual constraint $r(z,u,\dot z)=\dot z-f(z,u;\theta)=0$ Physical consistency is encoded as zero residual between measured rates and governing equations. Learn the zero-set of biochemical dynamics and reject rate-shuffled or direction-wrong samples.
Batch Fermentation Biomass-substrate-product dynamics $\frac{dX}{dt}=\frac{30.87XS}{(X+121.87)(S+105.4)}$ Biomass growth depends jointly on biomass and substrate through saturating kinetics. Verify whether biomass-rate data satisfy the learned fermentation growth law.
Batch Fermentation Substrate consumption law $\frac{dS}{dt}=-\frac{440.99XS}{(X+121.87)(S+105.4)}$ Substrate is consumed as biomass grows, with a rate linked to the same kinetic denominator. Detect unphysical trajectories where substrate decreases too slowly, too fast, or inconsistently with biomass growth.
Batch Fermentation Product formation law $\frac{dP}{dt}=\frac{73.65XS}{(X+121.87)(S+105.4)}$ Product accumulates as a coupled consequence of biomass growth and substrate conversion. Verify whether product-generation data lie on the same kinetic manifold as biomass and substrate dynamics.
Microbial Growth Monod growth law $\mu(S)=\mu_{\max}\frac{S}{K_S+S}$ Specific growth rate saturates with substrate concentration. Learn whether observed growth rates follow substrate-limited biological kinetics rather than arbitrary nonlinear trends.
Enzyme Kinetics Michaelis-Menten law $v(S)=\frac{V_{\max}S}{K_M+S}$ Enzyme-catalyzed reaction rates saturate as substrate becomes abundant. Verify whether enzyme-rate data lie on the Michaelis-Menten kinetic manifold.
Population Dynamics Logistic growth $\frac{dX}{dt}=rX\left(1-\frac{X}{K}\right)$ Biomass grows exponentially at low density but saturates due to carrying capacity. Detect unphysical biomass curves that violate growth saturation constraints.
Bioprocess Dynamics Biomass inhibition $\mu(X,S)=\mu_{\max}\frac{S}{K_S+S}\left(1-\frac{X}{K_\mu+X}\right)$ High biomass can inhibit further growth due to crowding, toxicity, or resource limitations. Test whether M-Layer learns full inhibitory dynamics instead of only a Monod substrate shell.
Temperature-Dependent Kinetics Arrhenius activation $k(T)=A\exp\left(-\frac{E_A}{RT}\right)$ Reaction and growth rates depend exponentially on temperature. Verify whether temperature-dependent process data obey thermodynamic rate scaling.
Bioreactor Control Chemostat biomass balance $\dot X=(\mu(S)-D)X$ Biomass changes according to biological growth minus dilution washout. Learn the admissible manifold of continuous bioreactor operation under dilution control.
Bioreactor Control Chemostat substrate balance $\dot S=D(S_{\mathrm{in}}-S)-\frac{1}{Y_{X/S}}\mu(S)X$ Substrate concentration changes due to feed, outflow, and biological consumption. Verify whether feed-substrate-growth data are mutually consistent.
Fed-Batch Processing Volume balance $\dot V=F$ Reactor volume increases according to the feed rate. Detect generated fed-batch trajectories with inconsistent feed-volume dynamics.
Fed-Batch Processing Dilution relation $D=\frac{F}{V}$ Dilution rate is determined by feed rate and reactor volume. Verify whether control inputs and reactor states satisfy process-operating constraints.
Product Formation Luedeking-Piret law $\dot P=\alpha\mu X+\beta X-DP$ Product formation can be growth-associated and non-growth-associated, with dilution loss in continuous systems. Learn whether product trajectories are consistent with biomass growth and process dilution.
Yield Constraints Biomass yield relation $-\dot S=\frac{1}{Y_{X/S}}\dot X$ Substrate consumption and biomass formation are linked by a yield coefficient. Test whether M-Layer learns stoichiometric yield constraints rather than only fitting each variable separately.
Yield Constraints Product yield relation $\dot P=Y_{P/S}(-\dot S)$ Product generation is constrained by substrate conversion. Verify whether product formation is physically coupled to substrate use.
Metabolic Networks Stoichiometric flux balance $\dot c=Nv(c)$ Metabolite concentrations evolve according to stoichiometric reaction networks. Learn whether metabolic trajectories satisfy network-level mass-conservation constraints.
Metabolic Networks Steady-state flux constraint $Nv=0$ At metabolic steady state, internal metabolite accumulation vanishes. Verify whether inferred flux vectors lie in the stoichiometric null space.
Bioreactor Thermodynamics Reactor energy balance $\rho C_pV\dot T=-\Delta H_rVr+UA(T_c-T)$ Temperature evolves due to reaction heat and heat exchange with the coolant. Verify whether concentration and temperature trajectories obey coupled mass-energy constraints.
Oxygen-Limited Bioprocesses Oxygen transfer law $\dot C_O=k_La(C_O^*-C_O)-q_OX$ Dissolved oxygen changes due to gas-liquid transfer and cellular uptake. Detect oxygen trajectories that violate transfer-uptake balance.
Maintenance Metabolism Pirt relation $q_S=\frac{\mu}{Y_{X/S}}+m_S$ Substrate uptake supports both growth and cellular maintenance. Learn whether substrate consumption decomposes into growth-associated and maintenance-associated components.
Process Optimization Productivity objective $\mathcal{P}=\frac{P(t_f)}{t_f}$ Bioprocess performance depends on final product concentration per unit time. Verify whether optimized trajectories remain dynamically feasible while improving productivity.
Sustainable Bioprocessing Carbon conversion efficiency $\eta_C=\frac{\text{carbon in product}}{\text{carbon in substrate}}$ Sustainable bioprocesses should convert substrate carbon efficiently into desired product. Learn whether generated process designs respect both kinetic feasibility and carbon-efficiency constraints.
Dynamic Equation Discovery Symbolic kinetic recovery $\dot z=f_{\mathrm{symbolic}}(z)$ Interpretable process models aim to recover compact closed-form kinetic laws from dynamic data. Use M-Layer as a verifier of candidate symbolic equations learned from bioprocess time series.
M-Layer Bioprocess Verifier Learned physicality score $\Psi_\theta(z,u,\dot z)\approx 0$ if $\dot z=f(z,u)$ A low score means the point lies on the learned biochemical process manifold. Distinguish true bioprocess states from derivative-shuffled, norm-preserving, or yield-preserving fake samples.





















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.
  • 📍 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.
  • 📍 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











References