- Lectures starting at 12:15
- Last year’s lecture recordings are available on ELTE Teams
- What Every Computer Scientist Should Know About Floating-Point Arithmetic link
- A detailed description of machine number representation, floating-point numbers, rounding errors

**HW:**Read Preface + Ch. 1-3 of the textbook- Please read the reading materials! There will be no detailed “lecturing” at the classes, only discussion of questions, you may have.

- Check out the project gudelines at the projects page ! Please note the deadlines!
- Questions, requests, etc. concerning the class:
**contact: szamszimmsc(at)gmail.com**

- Moore’s law of cameras: LSST 3.2 gigapixel camera photos released
- Christian Szegedy PhD, Research Scientist at Google Research lecture at SZTAKI Szept 15. 17:00 : Towards the Automatic Mathematician zoom
- Marcus Du Sautoy: The Creativity Code book on AI and math
**HW:**Read Ch. 4-5 of the textbook

- Check out the new course project webpage
- Please
**LOGIN TO KOOPLEX AND CHECK IF YOU SEE THE COURSE!**There is a**Tutorial**for newbies. - Computer language evolution: Machine code, assembly, C, C++, Python … next step maybe AI can write programs? OpenAI Codex and GitHub Copilot
- True (?) random number generator hardware or lava lamps(?!)
- DRAM based random number generators: https://arxiv.org/abs/2201.01385 and https://arxiv.org/pdf/1808.02068.pdf
- A good software RNG, the Mersenne twister
- Random numbers and Python
- What problems a bad RNG may cause: Random number generator attack
**Quantum computers as RNG: Quantum Supremacy (2019)***Quantum supremacy is the point at which quantum computers can solve problems that are practically unsolvable for “classical” (non-quantum) computers to complete in any reasonable timeframe. It is generally believed that at least 49 qubits are required to cross the quantum supremacy line.**Google’s ‘Sycamore’ quantum computer was able to achieve “quantum supremacy” — solving a complex problem that would otherwise be impossible for a classical computer to solve in its lifetime — in just three minutes and 20 seconds, compared to the estimated 10,000 years it would take the world’s most advanced classical computer, Summit.**To demonstrate quantum supremacy, we compare our quantum processor against state-of-the-art classical computers in the task of sampling the output of a***pseudorandom quantum circuit.***Due to quantum interference, the probability distribution of the bitstrings resembles a speckled intensity pattern produced by light interference in laser scatter, such that some bitstrings are much more likely to occur than others. Classically computing this probability distribution becomes exponentially more difficult as the number of qubits (width) and number of gate cycles (depth) grows.*- some explanation
- some more explanation
- original paper

- Liu, F.M. et al. Quantum Design for Advanced Qubits.
**2021**arXiv preprint arXiv:2109.00994. - Zhu, Q., Cao, S., Chen, F., Chen, M.C., Chen, X., Chung, T.H., Deng, H., Du, Y., Fan, D., Gong, M. and Guo, C., 2021. Quantum Computational Advantage via 60-Qubit 24-Cycle Random Circuit Sampling. arXiv preprint arXiv:2109.03494.
**HW:**Read Ch. 6-7 of the textbook

- Riemann vs. Monte Carlo integral scaling notebook
- Neumann rejection method explanation and code
- Root finding summary slides (based on Numerical Recipes book)
- Root finding convergence, Newton fractal notebook and zooming movie
**HW:**Read Ch. 8-9. of the textbook

- Nice real example for
**many-body gravitational**system: First (possible) observation of (exo)planet in a triple star system - 2021
**Nobel prize**in physics for greenhouse effect, climate modeling and**complex systems**(spin glasses) summaries*Related:*DeepMind can predict rain (in Britain :-) from Doppler radar for the next hour: Ravuri, S., Lenc, K., Willson, M. et al. Skilful precipitation nowcasting using deep generative models of radar. Nature 597, 672–677 (2021). link

**Can we improve simple algorithms like multiplication?**- https://www.quantamagazine.org/mathematicians-discover-the-perfect-way-to-multiply-20190411/
- https://hal.archives-ouvertes.fr/hal-02070778/document
- Strassen method: https://en.wikipedia.org/wiki/Strassen_algorithm
- Coppersmith–Winograd algorithm: https://en.wikipedia.org/wiki/Coppersmith%E2%80%93Winograd_algorithm)

**Maximum likelihood**vs**Least squares**fit slide- … and notebook

**HW:**Read Ch. (9)-10. of the textbook

- Derivation for the fourth order Runge-Kutta method which follows a similar logic as the one for rk2 in Landau’s book (in 9.5.2.) is not that simple. This is a “simplified” derivation: https://www.researchgate.net/publication/49587610_A_Simplified_Derivation_and_Analysis_of_Fourth_Order_Runge_Kutta_Method
- Ordinary Differential Equations - explanation slides
- Predictor-Corrector Method explanation
- Energy conservation notebook

- Methods for
**differentiation and calculating gradients**, on emphasis on neural network weight optimization- gradient methods animated figure
- gradient methods described
- automatic differentiation, local annotated copy
- autodiff: https://justindomke.wordpress.com/2009/02/17/automatic-differentiation-the-most-criminally-underused-tool-in-the-potential-machine-learning-toolbox/
- automatic differentiation short explanation

- Visualization of multidimensional minimization Loss Landscape
- Machine learning as a useful tool for cosmology studies:
- Francisco Villaescusa-Navarro et al. : The CAMELS project: Cosmology and Astrophysics with MachinE Learning Simulations
- Sedaghat, N., Romaniello, M., Carrick, J.E. and Pineau, F.X., 2021. Machines learn to infer stellar parameters just by looking at a large number of spectra. ]Monthly Notices of the Royal Astronomical Society, 501(4), pp.6026-6041.](https://arxiv.org/pdf/2009.12872.pdf) (using variational autoencoders)
- Machine learning to analyse gravitational lensing observations
- Machine learning to replace N-body simulations

- Machine learning in physics review: Carleo, G., Cirac, I., Cranmer, K., Daudet, L., Schuld, M., Tishby, N., Vogt-Maranto, L. and Zdeborová, L., 2019. Machine learning and the physical sciences. Reviews of Modern Physics, 91(4), p.045002. arxiv
- Review on “Physics Informed Neural Networks” :
- Karniadakis, G.E., Kevrekidis, I.G., Lu, L., Perdikaris, P., Wang, S. and Yang, L., 2021. Physics-informed machine learning. Nature Reviews Physics, 3(6), pp.422-440.
**HW:**Read Ch. 10-11

- Fourier analysis
- Viusal interactive explanation of Fourier analysis
- IEEE Top 10 algorithms of the 20th century. Some details
- gravitational wave data analysis from LIGO Open Science Center
- “FFT” on a sphere: Cosmic Microwave Background radiation analysis. Detailed example notebooks
- Nyquist sampling theorem/ compressive sensing

- Recent insight: Relation between wavelet transformation and tensor network decomposition
**HW:**Read Ch. 12

- Fermi-Ulam-Pasta-Tsingou system and original paper
- Mary Tsingou’s contribution to the Fermi-Ulam-Pasta study
- Logistic map (nice animations)
- In 1975, Dr. Feigenbaum, using the small HP-65 calculator discovered that the ratio of the difference between the values at which such successive period-doubling bifurcations occur tends to a constant of around 4.6692.
- Stretch-and-fold chaos and the horseshoe map
- Coupled chaotic maps
- Chaotic double pendulum notebook
- Lorenz attractor
- Dimensionality of the underlying dynamics: Taken’s theorem and the Grassberger-Procaccia algorithm
- Chaos in dripping faucet time series
- Chaotic attractors
- Related question: Weather vs. climate. See 2021
**Nobel prize**in physics and also the ongoing COP26 - Drótos, G., Bódai, T. and Tél, T., 2017. On the importance of the convergence to climate attractors. The European Physical Journal Special Topics, 226(9), pp.2031-2038. link

- Related question: Weather vs. climate. See 2021
- Chaotic embedding method explained
- News: related to Grassberger-Procaccia algorithm and Takens theorem:
- ‘AI Copernicus ‘discovers’ that Earth orbits the Sun’ review
- paper
- Recovering Hamiltonian with machine learning link
- Asseman, A., Kornuta, T. and Ozcan, A., 2018. Learning beyond simulated physics.
- real experimental data to train machine learning

- Lotka-Volterra population dynamics model and original hare-lynx time series
- not only animal population
- COVID-19 “agent based modeling” vs. SIR model
**predator prey like systems in chemistry**:- Briggs–Rauscher oscillating reaction (nice video)
- Belousov–Zhabotinsky reaction (with simulation demo) and experiment video1 , video2
- Examples of more complex systems: Citric acid cycle , Metabolic pathway, “Google map” of metabolic pathways

**Chaotic simulation precision**- Sympletic integrators for nonlinear Hamiltonian systems: https://en.wikipedia.org/wiki/Symplectic_integrator
- Corless, R.M., 1994. What good are numerical simulations of chaotic dynamical systems?. Computers & Mathematics with Applications, 28(10-12), pp.107-121. link
- Li, X. and Liao, S., 2018. Clean numerical simulation: a new strategy to obtain reliable solutions of chaotic dynamic systems. Applied Mathematics and Mechanics, 39(11), pp.1529-1546. link
- Boekholt, T.C.N., Portegies Zwart, S.F. and Valtonen, M., 2020. Gargantuan chaotic gravitational three-body systems and their irreversibility to the Planck length. Monthly Notices of the Royal Astronomical Society, 493(3), pp.3932-3937. link :
*“… using the accurate and precise N-body code Brutus, which goes beyond standard double-precision arithmetic … three massive black holes with zero total angular momentum, we conclude that up to five percent of such triples would require an accuracy of smaller than the Planck length in order to produce a time-reversible solution”*

**HW:**Read Ch. 13-14

**Fractals:**- Mandelbrot set
- Onsager prize, Tamas Vicsek. He had important role in the development of the fractal field.
- fractal dimension
- box counting method
- fractal GPS path

- example fractal dimensions
- DNA fractal Peano-Hilbert globule article1 , article2
- Heart rate fluctuation dynamics fractal
- Arnold’s cat map
- Easy 2D experiment: camera watching screen with camera’s recording
- fractal antenna
- genetic algorithm related to Barnsley’s fern and also evolutionary models
- Some fun with Diffusion Limited Aggregation
**Cellular automata:**- 1D
- NKS
- Wolfram Physics Project - Cellular Automata as building blocks of natural laws
- Golly
- Neumann universal constructor with a surprise
- Digits of Pi
- Gerard ‘t Hooft’s idea
- on general graph: Kauffman’s NK boolean network and some analysis
**High performance computing:**- Top 500
**HW:**Read Ch. 15-16

- News 2021:
- Planet 9 discovered? Popular article
- IBM announces a new 127-qubit quantum processor

- Empowering simulations with “physics informed” neural nets
- Karniadakis, G.E., Kevrekidis, I.G., Lu, L., Perdikaris, P., Wang, S. and Yang, L., 2021. Physics-informed machine learning. Nature Reviews Physics, 3(6), pp.422-440.
- Nvidia SimNet toolkit

**Thermodynamic simulations, Ising model:**- Simplest model of phase transition: Ising model, exact 2D solution, Onsager 1944 and a real life 2D ferromagnet (2018)
- Faster dynamics, spin clusters: Swendsen-Wang algorithm, visualization
- Scaling at the critical point, renormalization demonstration
- Metropolis algorithm and simulated annealing
- Hopfield model
- Neural nets have similar complex energy landscape: visualizations , article
- Numerical optimization and stat phys.: Moore & Mertens: The Nature of Computation , short summary
- NP-hardness concept, list of problems ; protein folding is NP-hard? or not
- One of the 2021
**Nobel prize**in physics: Giorgio Parisi for (spin glasses)

**Molecular dynamics**- long range force cutoff, Ewald: link link2
- initial transient. Start with Maxwell distr. but since the coordinate-velocity correlations are not correct, temperature rescaling needed: link
- protein folding link
- energy landscape: link
- badly folded proteins: prions
- protein folding with machine learning: alphaFold
- AlphaFold2 paper and a blog explaining the overall method.
- hoomd-blue open source molecular dynamics simulator program
- argon with Hoomd
- two/three component mixing with Hoomd
- interesting problem: tetrahedron packing theoretical limit and hoomd simulation

**Partial differential equations**- classification and boundary conditions and explanation of names
- Green’s Function explained with linear algebra analogy: @Artem at math.stackexchange
- Finite element method and comparision to Finite difference method link
- FTCS and Crank-Nicolson methods as stepping examples
- von Neumann stability analysis link
- openFOAM open source computational fluid dynamics program
- FEniCS open source finite element numerical solver for PDEs
- Meep open source finite-difference time-domain PDE solver
- Python 3D FDTD Simulator another open source finite-difference time-domain PDE solver (with PyTorch backend)

**HW:**Read Ch. 17 (and if you are interested, the rest of the book)**Prepare for the presentations starting next week!**