Algorithmic Physics: Computation in Nature

Updated 13 November 2025

Algorithmic physics is an interdisciplinary framework that models physical phenomena as computational and algorithmic processes.
It bridges theoretical computer science, information theory, and machine learning to reinterpret traditional physical laws through concrete algorithmic analogies.
Applications include simulating oxidation kinetics, martensitic transformations, and quantum measurements using data structure and complexity insights.

Algorithmic physics is a research paradigm and interdisciplinary framework in which physical phenomena are modeled, analyzed, and, in some cases, fundamentally understood in terms of algorithmic and computational structures. This perspective fuses ideas from theoretical computer science, information theory, optimization, numerical analysis, machine learning, and foundational physics to recast both the mathematical laws of nature and their empirical manifestations as either explicit or implicit algorithms. The scope of algorithmic physics spans from concrete implementations—where natural processes are simulated or inferred using computational algorithms—to foundational claims that the evolution of physical systems is itself intrinsically computable or governed by principles of algorithmic probability, informational simplicity, or computational complexity.

1. Core Principles and Foundational Models

The foundational principles of algorithmic physics rest on two broad classes of claims: (1) Nature can be effectively modeled or even viewed “as a computer,” implementing algorithms—either explicitly, in the sense of dynamical systems governed by algorithmic rules, or implicitly, as the solution to optimization or inference problems with clear computational structure (Pop et al., 2012); (2) Physical law emerges as an epiphenomenon of algorithmic or information-theoretic principles, with observable regularities governed by processes such as universal induction, minimization of computational resources, or selection for informational simplicity (Mueller, 2017, Mueller, 3 Dec 2024, Sienicki, 20 Jan 2025).

Observable processes (e.g., metallurgy oxidation, shape-memory effect, quantum measurement) may be mapped to or characterized by the algorithmic complexity of their time-evolution laws, e.g., identifying empirical kinetic laws as analogs of algorithmic time complexity classes: linear (O(n)), quadratic (O(n²)), or exponential (O(2ⁿ)) (Pop et al., 2012). Data-structure abstractions such as stacks and queues find physical analogs, testifying to deep structural similarities between information processing and physical dynamics.

Central to “algorithmic idealism” (Mueller, 3 Dec 2024, Sienicki, 20 Jan 2025, Mueller, 2017) is the idea that observer-centric notions of physical law—modeled by conditional algorithmic probability (Solomonoff induction and Kolmogorov complexity)—can reproduce the apparent objectivity and regularity of the external world, dissolve cosmological paradoxes (e.g., the Boltzmann brain problem), and account for core features of quantum theory as emergent from fundamental computational constraints.

2. Algorithmic Modeling of Physical Processes

Physical systems are frequently cast as discrete or continuous algorithms, with time, space, or physical observables mapped directly to computational notions such as input size, running time, memory, or algorithmic resource constraints (Pop et al., 2012). Case studies include:

Oxidation kinetics: Classical growth laws (parabolic, logarithmic, linear) are mapped to O(n²), O(2ⁿ), or O(n) algorithmic time complexity, illuminating which physical processes manifest as efficient or inefficient computations.
Martensitic transformations in alloys: The reversible stacking and “popping” of crystal variants is rigorously described as a Last-In-First-Out stack process, exactly mirroring the stack data structure of computer science.
Optimization algorithms: Canonical algorithms such as gradient descent, Nesterov’s acceleration, and Newton’s method are reinterpreted as damped mechanical oscillators, allowing convergence rates and momentum parameters to be derived via physical analogies and Lagrangian mechanics (Yang et al., 2016).

A summary of the mapping between physical system, algorithmic abstraction, and computational complexity is given below:

Physical Process	Algorithmic Model	Complexity or Data Structure
Parabolic oxide growth	Quadratic algorithm	O(n²) time
Shape memory LIFO	Stack	LIFO stack
Gradient descent	Overdamped oscillator	Energy decay/Lyapunov
Newton’s method	Anisotropic damping	Position-dependent flow

3. Algorithmic Information Approaches to Physical Law

The algorithmic information approach postulates that the laws of physics may themselves be products of algorithmic probability and the preference for informationally simple continuations, as expressed in Solomonoff induction and Kolmogorov complexity (Mueller, 2017, Mueller, 3 Dec 2024, Sienicki, 20 Jan 2025).

Formally, let U be a universal Turing machine and K_U(x) the prefix Kolmogorov complexity of a binary string x. The universal prior or Solomonoff measure M_U(x) is defined by summing over all minimal programs that output x as a prefix: $M_U(x) = \sum_{p: U(p)=x*} 2^{-|p|}$ The conditional probability of “seeing” a next state a given an observer state x is

$P_U(a|x) \approx \frac{M_U(xa)}{M_U(x)}$

The main postulate asserts that, fundamentally, the probability of an observer experiencing the next bit a in state x is exactly this universal a priori probability. As shown in (Mueller, 2017), this framework recovers (with high algorithmic probability) the apparent emergence of a simple, computable, probabilistic world, capable of reproducing quantum-like phenomena and resolving key cosmological paradoxes without referencing objective reality as primitive.

Extensions such as “Algorithmic Idealism” (Mueller, 3 Dec 2024) and the “Algorithmic State” formulation of quantum mechanics (Sienicki, 20 Jan 2025) introduce further informational constructs such as utility, Bayesian updating for measurement, and simulation/identity equivalence, tying together agent-based decision making and quantum observation under a unified computational theory.

4. Computational Physics as Algorithm

A key aspect of algorithmic physics is the engineering of algorithms whose structure closely reflects the physical constraints, symmetries, and invariants of the underlying system. This approach goes beyond generic numerical simulation, embedding domain-specific structure directly in both classical and machine-learning algorithms.

Physical simulation via integrators: Methods such as the explicit Euler, midpoint, Feynman, and Runge-Kutta integrators are systematically compared for real-time softbody simulation, with the Feynman (midpoint-like) algorithm achieving better stability and almost second-order accuracy at modest computational overhead (0906.3074).
Cluster expansions and efficient counting: In low-temperature statistical physics (Potts, hard-core models), a union of Pirogov-Sinai contour theory and Barvinok’s Taylor-series approach enables deterministic FPTAS algorithms for partition functions and efficient sampling (Helmuth et al., 2018), filling longstanding gaps where classical MCMC approaches are torpidly mixing.
Message-passing and complexity boundaries: Statistical-physics intuition and rigorous mathematics (e.g., Parisi’s replica symmetry breaking, Talagrand’s Parisi formula) have led to precise algorithmic phase diagrams for random optimization, e.g. k-SAT, community detection, and spin glasses (Gamarnik, 25 Jan 2025). Key insights are formalized as the Overlap-Gap Property (OGP), which establishes concrete complexity barriers above certain thresholds.
High-performance algebraic approaches: For quantum gravity and cosmology, specialized data structures (e.g., FastBitset), low-level algorithmic routines, and massively parallel computation provide order-of-magnitude improvements in computing causal set actions, geodesic distances, and vacuum-selection probabilities in string landscape models (Cunningham, 2018).

5. Data-Driven, ML, and AI-Discovered Physics

Recent developments in physics-aware machine learning and “algorithmic physics” frameworks have integrated physical laws or symmetries as inductive biases—or as explicit architectural constraints—within neural networks, producing models that are both physically plausible and computationally efficient (Tong, 20 Jun 2024).

Physics-embedded neural systems: Architectures such as symplectic Taylor networks (for separable Hamiltonian systems), nonseparable symplectic nets, RoeNet (a learned Roe solver for hyperbolic conservation laws), and neural vortex methods for incompressible fluid flow all embed classical algorithms or invariants into neural networks. This ensures preservation of energy, momentum, or other conserved quantities, and leads to robust performance even with small datasets, noisy inputs, or long extrapolation (Tong, 20 Jun 2024).
Universal physics simulation by conditional generative models: Diffusion-transformer frameworks treat steady-state solution generation as a conditional generative process, using boundary conditions to directly synthesize physical solutions without a priori equation encoding or sequential time-stepping. This supports AI-driven discovery of physical laws via learned internal representations (as revealed by Layer-wise Relevance Propagation identifying divergence-free constraints or emergent curl laws in electromagnetics) (Camburn, 13 Jul 2025).
Mesh-free variational learning in micromagnetism: By recasting variational bounds (Brown’s energy bounds) on a finite computational domain and rigorously enforcing hard constraints via Cayley transforms (unit norm) and R-functions (boundary conditions), fast mesh-free PINNs and Extreme Learning Machines achieve high accuracy and competitive runtime in 3D micromagnetic simulation, including hysteresis and energy minimization, compared to traditional FEM/BEM approaches (Schaffer et al., 19 Sep 2024).

6. Quantum and Foundational Algorithmic Frameworks

Quantum experimental design, measurement, and foundational issues have been recast within algorithmic frameworks:

Quantum Algorithmic Measurement (QUALMs): Quantum experiments are formalized as quantum circuits (or interactive protocols) with specified query- and gate-complexity; exponential coherent-incoherent separations have been established in tasks such as time-translation symmetry testing and unitary symmetry-class identification, providing a computational-complexity language for quantifying “algorithmic advantages” in experimental physics (Aharonov et al., 2021).
Algorithmic foundations of quantum theory: “Algorithmic State” quantum mechanics derives measurement, entanglement, and Born probabilities via Bayesian updating over algorithmic priors and utility-optimized observer transitions, eliminating the need for independent collapse postulates and yielding conservation laws via reward-function invariance (Sienicki, 20 Jan 2025).

7. Statistical Inverse Problems and Resolution Limits

The algorithmic perspective has sharpened the statistical understanding of physics-limited inference tasks, such as optical imaging and resolution limits:

Diffraction limit as statistical/algorithmic phase transition: The ability to resolve point sources below the Abbe limit is shown to exhibit a sharp transition from polynomial to exponential sample complexity, with practical separation thresholds strictly above classical Abbe/Rayleigh bounds. Algorithms based on moment methods, convex-optimization, tensor decomposition, and Fourier analysis achieve efficient recovery above the threshold, while below it, no algorithm is feasible with sub-exponential data (Chen et al., 2020).

Model	Key Complexity/Threshold	Feasible Algorithm Regime
Diffraction imaging	∆ ≈ 1.155 πσ (lower bound)	Poly(N, 1/∆, ..) if ∆ > 1.53 πσ
k-SAT	α < α_s(k) ~ 2^k ln2	BP/AMP up to “easy” threshold
Spin glass (SK, p-spin)	OGP/no-gap = efficient; gap = hard	AMP state evolution (dense), BP

8. Future Directions and Open Questions

Algorithmic physics continues to expand into:

Reliable automatic physics discovery, with architectures that identify new invariants or governing equations from empirical data, without pre-encoded PDE structure (Camburn, 13 Jul 2025).
Formal classification of algorithmic complexity classes for physical processes—“are there lower bounds on physical time mapped to minimal computational operations?” (Pop et al., 2012).
Extension of mesh-free, constraint-embedded machine learning to polycrystalline and multiphysics domains, and the solution of previously intractable large-scale variational problems (Schaffer et al., 19 Sep 2024).
Deepening foundational links between observer-centric algorithmic probability and the objective emergence of physical law and cosmological structure (Mueller, 2017, Mueller, 3 Dec 2024).
Algorithmic experiment design as an explicit field, optimizing sample and gate complexity for quantum and classical experiments (Aharonov et al., 2021).

The algorithmic physics paradigm thus encompasses both concrete, efficient mapping from physical laws to computation and deeper conceptual frameworks identifying computation as a candidate substrate or organizing principle of physical reality.