The following books are good text books and reference books

Math

Dynamical systems

Introductory textbook on differential equations

  • Arnold, V. I. (1992). Ordinary Differential Equations. Springer.

Introductory textbooks on nonlinear dynamical systems:

  • Strogatz, S. H. (2018). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press.

More advanced textbooks on bifurcations:

  • Kuznetsov, Y. A. (2013). Elements of Applied Bifurcation Theory. Springer.
  • Guckenheimer, J., & Holmes, P. (2013). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.

Nonequilibrium pattern formation:

  • Cross, M. C., & Hohenberg, P. C. (1993). Pattern formation outside of equilibrium. Reviews of Modern Physics, 65(3), 851.
  • 欧阳颀，非线性与斑图动力学导论，北京大学出版社，2006

Probability theory and Stochastic process

Probability theory

  • A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, 4th edition, McGraw-Hill, 2002.
  • S. M. Ross, Introduction to Probability Models, 11th edition, Academic Press, 2014.

Handbooks and applications of stochastic processes

  • Gardner, C. W. (2009). Stochastic Methods: A Handbook for the Natural and Social Sciences. Springer.
  • van Kampen, N. G. (2007). Stochastic Processes in Physics and Chemistry. North Holland.
  • H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, Springer, 1996.
  • Bernt Oksendal, Stochastic Differential Equations: An Introduction with Applications, 6th edition, Springer, 2003.
  • Simo Särkkä and Arno Solin, Applied Stochastic Differential Equations, Cambridge University Press, 2019.

Group theory

Wu-Ki Tung’s book is a classic and emphasizes the physical intuition behind group theory. It covers a wide range of topics from the basics of group theory to applications in physics. A. Zee’s book is comprehensive and written with a lot of humor, anecdotes and historical context, making it an enjoyable read. Though it sometimes breaks the flow of the material. Peter Woit’s book is more mathematically rigorous.

  • Wu-Ki Tung, Group theory in physics, World Scientific, 1985.
  • A. Zee, Group theory in a nutshell for physicists, Princeton University Press, 2016.
  • Ashok Das and Susumu Okubo, Group Theory for Physicists, World Scientific, 2010.
  • Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd edition, Springer, 2015.
  • Peter Woit, Quantum Theory, Groups and Representations: An Introduction, Springer, 2017.

Random matrix theory

Random matrix theory (RMT) is a very powerful and versatile tool for understanding the behavior of large-scale networks. For example, the behavior of random recurrent neural networks (RNNs).

  • Marc Portters and Jean-Philippe Bouchaud, “A First Course in Random Matrix Theory: For Physicists, Engineers and Data Scientists”, Cambridge University Press, 2022.
  • Terence Tao, “Topics in Random Matrix Theory”, American Mathematical Society, 2012
  • Giacomo Livan, Marcel Novaes and Pierpaolo Vivo, “Introduction to Random Matrices: Theory and Practice”, Springer, 2018.
  • Madan LalMehta, “Random Matrices”, Elsevier, 2004.
  • Lloyd N. Trefethen and Mark Embree, Spectral and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press, 2005.

Perturbation and Asymptotic Analysis

  • J. G. Simmonds and J. E. Mann Jr., A first look at perturbation theory, Dover publication Inc, 2nd ed, 1986

Variational methods

  • I. M. Gelfand and S. V. Fomin, Calculus of Variations, Dover Publications Inc, 2000.

Topology and Geometry

  • Tristan Needham, Visual differential geometry and forms: a mathematical drama in five acts, Princeton University Press, 2021.
  • John M. Lee, Introduction to Topological Manifolds, 2nd edition, Springer, 2010.
  • John M. Lee, Introduction to Smooth Manifolds, 2nd edition, Springer, 2012.

Statistics

  • T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd edition, Springer, 2009.
  • G. Casella and R. L. Berger, Statistical Inference, 2nd edition, Duxbury, 2001.
  • Larry Wasserman, All of Statistics: A Concise Course in Statistical Inference, Springer, 2004.
  • H. Kantz, A. J. Schreiber, and T. P. Sejnowski, Nonlinear Time Series Analysis, 2nd edition, Cambridge University Press, 2004.
  • R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, 1982.

    Physics

    Statistical Physics

    Basic level introductory statistical physics books.

  • W. Greiner, L. Neise, and H. Stöcker, Thermodynamics and Statistical Mechanics, Springer, 1995.
  • L. Landau and E. M. Lifshitz, Statistical Physics, 3rd edition, Part 1, Butterworth-Heinemann, 1980.
  • Kerson Huang, Statistical Mechanics, 2nd edition, Wiley, 1987.
  • Mehran Kardar, Statistical Physics of Particles, Cambridge University Press, 2007

More advanced books, statistical field theory which will cover tools for critical phenomena such as the Renormalization group theory.

  • Mehran Kardar, Statistical Physics of Fields, Cambridge University Press, 2007.
  • S-K, Ma, Modern Theory of Critical Phenomena, Westview Press, 2000.
  • Nigel Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, CRC Press, 2018.
  • J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newman, The Theory of Critical Phenomena: An Introduction to the Renormalization Group, Oxford University Press, 1992.
  • Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford University Press, 2002.
  • J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, 1996.

Spin glasses and related

A popular book on spin glasses and complexity, very readable for outsiders. You can get a sense of the history of the field

  • Daniel L. Stein and Charles M. Newman, Spin Glasses and Complexity, Princeton University Press, 2013.

More advanced books on spin glasses and replica theory

  • M. Mézard, G. Parisi, and M. A. Virasoro, Spin Glass Theory and Beyond, World Scientific, 1987.
  • H. Nishimori, Statistical Physics of Spin Glasses and Information Processing: An Introduction, Oxford University Press, 2001.
  • V. Dotsenko, Introduction to the Replica Theory of Disordered Statistical Systems, Cambridge University Press, 1995.

Field theory

To get some exposure to quantum field theory

Miscellaneous

Phillips W. Anderson has a memorior published in 1994, “More and Different: Notes from a Thoughtful Curmudgeon”, World Scientific. It has many interesting stories about the development of condensed matter physics in the second half of the 20th century.

  • P. W. Anderson, More and Different: Notes from a Thoughtful Curmudgeon, World Scientific, 1994.

Back of the envelope physics

  • 赵凯华，定性与半定量物理学，高等出版社，2008
  • A. Zee, Fly by Night Physics: How Physicists Use the Backs of Envelopes to Solve Problems, Princeton University Press, 2020. It has a Chinese translation:
  • 徐一鸿，物理夜航船：直觉与估算，高等教育出版社，2021

Computer Science/ Machine Learning

  • Bishop, C. M. (2006). Pattern Recognition and Machine Learning. Springer.
  • D. G. Luenberger, Optimization by Vector Space Methods, Wiley, 1997.
  • S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.

Introduction to deep learning

  • Ian Goodfellow, Yoshua Bengio, and Aaron Courville, Deep Learning, MIT Press, 2016. (freely available online at https://www.deeplearningbook.org/)

PyTorch based deep learning book

  • Eli Stevens and Luca Antiga, Deep Learning with PyTorch, Manning Publications, 2020.

Information theory

Best introductory book on information theory, very readable.

A standard reference book on information theory.

  • Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, 2nd edition, Wiley-Interscience, 2006.

  • David J.C. MacKay, Information Theory, Inference, and Learning Algorithms, Cambridge University Press, 2003.

    Neuroscience

    The ‘bible’ of neuroscience, which is very comprehensive but can be overwhelming for beginners.

  • Kandel ER, Koester JD, Mack SH 2021, Siegelbaum SA. Principles of Neural Science, 6th ed. McGraw-Hill, 2021.

The one from Liqun Luo is more friendly to beginners

  • Liqun Luo, Principles of Neurobiology, 2nd edition, 2020 Similarly, the one from Bear et al

and a recent textbook published by the Higher Education Press (Chinese)

  • 饶毅、梅林、段树民等，神经生物学导论， 高等教育出版社，2025

If you want to learn more about the design principles of neural systems, Sterling and Laughlin have a very nice book on the topic:

  • Peter Sterling, Simon Laughlin, “Principles of Neural Design”, MIT Press, 2015.

A dialogue style popular book on neural development is very readable:

  • Peter Robin Hiesinger,”The Self-Assembling Brain: How Neural Networks Grow Smarter”, Princeton University Press, 2021

Theoretical and computational neuroscience

  • Peter Dayan and L. F. Abbott, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, MIT Press, 2001.
  • J. Hertz, A. Krogh, and R. G. Palmer, Introduction to the Theory of Neural Computation, Addison-Wesley, 1991.

Statistcal physics approaches to neural networks

  • M. Helias, D. Dahmen, Statistical Field Theory for Neural Networks, Springer, 2020.
  • T. Geszti, Physical models of neural networks, Wold Scientific, 1990

Scientific writing

  • Hilary Glasman-Deal, Science Research Writing for Non-Native Speakers of English, 2nd edition, Imperial College Press, 2010.
  • E. B. White, The Elements of Style, 4th edition, Longman, 2000.
  • R. H. Fiske, To the Point: A Dictionary of Concise Writing, HarperCollins, 2001.
  • J. Scheff, Writing Science: How to Write Papers That Get Cited and Proposals That Get Funded, 2nd edition, Oxford University Press, 2013.

Science History and autobiography

Although Waston was critized by not giving enough credit to Rosalind Franklin, this book is still a very interesting read about the discovery of the DNA structure.

  • James D. Watson, The Double Helix: A Personal Account of the Discovery of the Structure of DNA, 1968.

Crick’s book is more reflective and philosophical about the nature of scientific discovery.

  • Francis Crick, What Mad Pursuit: A Personal View of Scientific Discovery, 1988.