Physics is full of equations that fit on a single line but resist every attempt at exact solution. The difficulty is rarely about the symbols themselves; it is about what happens when nonlinearity, many interacting bodies, or extreme scales make the math explode in complexity. Some of these equations, like the Navier-Stokes equations governing fluid flow, are so stubbornly unsolved that a million-dollar prize awaits anyone who can prove basic facts about their behavior. Others, like the quantum many-body problem, grow exponentially harder with every particle you add, straining even the largest supercomputers.
The Three-Body Problem
Newton’s law of gravity is simple enough when two objects are involved. Two bodies orbiting each other follow tidy ellipses, and a clean formula describes their motion for all time. Add a third body, and the situation changes dramatically. The gravitational three-body problem has been studied for centuries and remains one of the most recognizable “hard” equations in physics, not because the underlying force law is complicated, but because the interactions among three mutually attracting masses produce chaotic behavior that makes general analytical solutions impossible.1Astronomy & Astrophysics. Isles of regularity in a sea of chaos amid the gravitational three-body problem
Chaos here means extreme sensitivity to initial conditions. Nudge the starting position of one body by a tiny amount and the long-term trajectories diverge wildly. Researchers have developed statistical approaches based on the assumption that chaos fully mixes the possible states of the system, an idea rooted in the ergodic principle. Even so, “isles of regularity” persist within the chaotic sea, special configurations where the motion remains orderly.2Astronomy & Astrophysics. Isles of regularity in a sea of chaos amid the gravitational three-body problem Those islands are narrow and hard to find, which is why the three-body problem remains a benchmark for dynamical difficulty. The takeaway for a general reader is that just because a force law is simple does not mean the resulting motion will be predictable.
The Navier-Stokes Equations and Turbulence
The Navier-Stokes equations describe how fluids, everything from water in a pipe to air flowing over an airplane wing, move in response to pressure, viscosity, and external forces. We use them constantly in engineering, weather forecasting, and aerodynamics. And yet nobody has managed to prove, in the full three-dimensional case, whether solutions that start out smooth always stay smooth or could instead blow up into infinite values. That question is one of the seven Millennium Prize Problems posed by the Clay Mathematics Institute, with a one-million-dollar bounty attached.3Russian Mathematical Surveys. Sixth problem of the millennium: Navier-Stokes equations, existence and smoothness
What makes this so hard is the nonlinear feedback loop at the heart of the equations. The fluid’s velocity at any point depends on the pressure, which in turn depends on the velocity everywhere else. This circular dependency becomes fierce when the flow turns turbulent. Turbulence, the swirling, seemingly random motion you see in a fast river or smoke rising from a fire, involves energy cascading from large scales to tiny ones. Physicists have adapted techniques from quantum field theory, particularly renormalization group methods, to study the scaling patterns inside turbulent flows.4Physics Reports. Renormalization group theory for fluid and plasma turbulence These methods connect the large-scale statistical behavior of turbulence to the small-scale physics through fixed points in the mathematical structure.5PubMed. Turbulent compressible fluid: Renormalization group analysis, scaling regimes, and anomalous scaling of advected scalar fields
Progress has been made on related versions of the problem. Mathematical work has shown that proving smooth behavior for the periodic Navier-Stokes problem with an added forcing term would imply smooth behavior for both official formulations of the Millennium Prize question.6Analysis & PDE. Localisation and compactness properties of the Navier–Stokes global regularity problem But even this partial advance came with a catch: fully smooth solutions are not always possible in every category of starting data. The equations keep revealing new layers of difficulty.
The Quantum Many-Body Problem
In quantum mechanics, the behavior of a single particle is governed by the Schrödinger equation, which is usually manageable. The difficulty starts when you have many particles interacting with one another. The quantum state of a many-body system encodes correlations between every particle and every other particle, and the amount of information required to describe that state grows exponentially with the number of particles.7PubMed. Solving the quantum many-body problem with artificial neural networks For a handful of electrons, this is workable. For the hundreds of electrons in a moderately sized molecule, an exact solution is out of the question with any existing computer.
The Schrödinger equation for many interacting particles is perfectly well-defined. You can write it down. The problem is that solving it exactly remains intractable for most practical cases.8ChemRxiv. Beyond Exactness: Approximation Strategies to the Many-Body Schrödinger Equation for Modern Quantum Chemistry Applications Chemists and physicists have developed a rich toolkit of approximations, from density functional theory to coupled cluster methods, that trade exactness for tractability. Each method works well in certain regimes and poorly in others. The continuing quest is to find approximation strategies that are both computationally affordable and reliably accurate across a wider range of systems.
The Yang-Mills Mass Gap
Another Millennium Prize Problem involves Yang-Mills theory, the mathematical framework underlying the Standard Model of particle physics. The equations of Yang-Mills theory describe the strong and weak nuclear forces using fields that carry their own charge, unlike electromagnetism, where photons are electrically neutral. This self-interaction makes the math far more complex. The specific unsolved question asks whether the theory predicts a “mass gap,” a minimum positive energy for the particles described by the equations. Experimentally, the strong force does have a mass gap: its force carriers (gluons and the bound states they produce) are massive. But proving this rigorously from the Yang-Mills equations has not been accomplished to the satisfaction of the mathematical community.
Some researchers have proposed proofs using connections between gravity and gauge theories.9International Journal of Geometric Methods in Modern Physics. Gravity-Assisted Solution of the Mass Gap Problem for Pure Yang–Mills Fields These approaches are creative, but the problem remains officially open. Part of what makes it so difficult is that the strong force becomes stronger, not weaker, at low energies and large distances. Standard perturbative methods, where you treat the interaction as a small correction, break down entirely in this regime. Nonperturbative techniques are needed, and those are much harder to put on rigorous mathematical footing.
Quantum Gravity and the Problem of Time
General relativity describes gravity as the curvature of spacetime and works beautifully at large scales: planets, stars, galaxies. Quantum mechanics works beautifully at small scales: atoms, photons, subatomic particles. Combining the two into a single consistent framework, quantum gravity, has been one of the great open challenges in physics for nearly a century. The fundamental difficulty is that in general relativity, spacetime itself is a dynamic entity that bends and stretches, while quantum mechanics is typically formulated on a fixed background stage. Making spacetime both dynamic and quantum simultaneously leads to equations whose physical meaning is unclear.
One concrete manifestation of this tension is the “problem of time.” In ordinary quantum mechanics, time is an external parameter that ticks along while the system evolves. In general relativity, time is part of the geometry and can stretch and warp. When you try to write down the quantum version of Einstein’s equations, you get the Wheeler-DeWitt equation, and that equation has no explicit time variable at all. Research into this problem has explored whether a meaningful concept of time can be recovered by demanding that certain physical relationships (specifically, Ehrenfest’s equations linking quantum averages to classical behavior) hold.10Nuclear Physics B. Ehrenfest’s principle and the problem of time in quantum gravity The conclusion so far is that only a very restricted set of solutions to the Wheeler-DeWitt equation satisfies all the required conditions, leaving the problem open and perhaps suggesting that the equation itself needs modification or reinterpretation.
When Approximation Methods Break Down
Since exact solutions to most of these equations are unavailable, physicists rely heavily on approximation techniques. The most common is perturbation theory: start with a problem you can solve, then treat the hard part as a small correction and compute increasingly precise refinements. This works remarkably well in quantum electrodynamics, where it produces predictions accurate to ten decimal places. But it does not always work.
In quantum field theory, even “resummed” perturbative expansions, which reorganize the calculation to improve convergence, can fail at high energies. Research has demonstrated that in certain scalar field theories with a unique stable minimum, where perturbation theory should in principle capture all the physics, the resummed results can still violate unitarity (a fundamental requirement that probabilities add up to one) when you look at processes involving many particles at once.11Journal of High Energy Physics. The breakdown of resummed perturbation theory at high energies This means the approximation method itself is generating unphysical predictions, not because the physics is wrong, but because the mathematical tool is being pushed beyond its range of validity.
Another hard wall shows up in quantum Monte Carlo simulations. Monte Carlo methods try to estimate quantum properties by sampling random configurations, and they are among the most powerful numerical tools in physics. But for systems of fermions (electrons, quarks, and other particles with half-integer spin), these methods run into the “sign problem,” where positive and negative contributions nearly cancel each other, making the calculation noisy and unreliable. This is not just an inconvenience. The sign problem has been proven to be NP-hard, which in practical terms means that no general-purpose shortcut around it is likely to exist.12PubMed. Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations Solving it in general would simultaneously solve every problem in a famously difficult class of computational puzzles.
Nonlinear Wave Equations and Surprise Solvability
Not every nonlinear equation is hopelessly intractable. Some nonlinear wave equations turn out to have hidden mathematical structure that allows exact solutions, and studying these “integrable” cases helps physicists understand what makes other equations so much harder by comparison.
The Korteweg-de Vries equation, which describes shallow water waves, and the nonlinear Schrödinger equation, which appears in fiber optics and quantum mechanics, are the most famous examples. These equations support solitons, wave packets that travel without spreading, and can be solved exactly using a technique called the inverse scattering transform. Recent work has extended this framework into fractional versions of these equations, which describe wave transport in media where the underlying physics involves non-standard dispersion. These fractional integrable equations predict “superdispersive” transport of solitons that do not dissipate energy even in fractional media.13PubMed. Fractional Integrable Nonlinear Soliton Equations
Adding higher-order terms to these equations creates new complications and new solutions. The fifth-order extension of the nonlinear Schrödinger equation, for instance, introduces additional dispersion effects that primarily change how fast solitons travel, with complex downstream consequences for the wave dynamics. This extension also supports entirely new structures, like pairs of equal-sized solitons sitting on top of each other, that cannot exist in the standard version of the equation.14PubMed. Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms The broader lesson is that “hard” and “easy” in nonlinear physics often come down to whether a particular equation happens to have a hidden symmetry that unlocks exact solutions. When that symmetry is absent, even a superficially similar equation becomes vastly more difficult.
Plasma Physics and the Curse of Six Dimensions
Plasmas, the hot ionized gases found in stars, fusion reactors, and space environments, are governed by the Vlasov-Maxwell equations, which track how charged particles distribute themselves across both position and velocity. Because each particle has three position coordinates and three velocity coordinates, the full problem lives in six-dimensional phase space. Simulating this accurately is enormously expensive. Continuum Vlasov simulations can model plasma behavior with high accuracy, but the computational cost scales brutally with the resolution you need in velocity space.15Journal of Computational Physics. A parallel low-rank solver for the six-dimensional Vlasov–Maxwell equations
This is one area where quantum computing has been proposed as a potential game-changer. Classical algorithms for the Vlasov-Maxwell system scale roughly as the sixth power of the number of grid points multiplied by the square of the simulation time. Quantum algorithms could in principle reduce this dramatically, with computational cost scaling only logarithmically with the number of grid points rather than as a power law.16Journal of Plasma Physics. Quantum calculation for two-stream instability and advection test of Vlasov–Maxwell equations Whether real quantum hardware will achieve this speedup anytime soon remains an open question, but the theoretical scaling advantage is striking enough to motivate serious development.
Numerical Relativity and Black Hole Mergers
Einstein’s field equations of general relativity are a system of ten coupled nonlinear partial differential equations. For simple, highly symmetric situations, like a non-rotating black hole or an idealized expanding universe, exact solutions exist and are celebrated results. For anything messier, such as two black holes spiraling into each other, you need numerical relativity: discretizing the equations and solving them on powerful computers step by step.
Simulating binary black hole mergers is one of the triumphs of computational physics. These simulations were essential for interpreting the gravitational wave signals detected by LIGO starting in 2015. But even with decades of development, the numerical methods involve delicate choices. When the black holes are immersed in magnetized plasma, as supermassive black holes in galactic centers would be, the electromagnetic emission during the merger has to be computed alongside the gravitational dynamics. Researchers have found that although dual jets form during such mergers, the jet luminosity is roughly a hundred times smaller than the total electromagnetic output, which is dominated by a different radiation pattern.17The Astrophysical Journal. Accurate Simulations of Binary Black Hole Mergers in Force-free Electrodynamics Getting these details right requires careful numerical treatment of the force-free condition for the plasma, which is itself an approximation layered on top of the already-difficult Einstein equations.
Neural Networks as Equation Solvers
One of the more unexpected developments in recent years is the use of neural networks to tackle hard physics equations directly. Physics-informed neural networks, known as PINNs, are trained to satisfy a differential equation rather than fit data in the traditional machine-learning sense. The neural network proposes a solution, and its loss function penalizes violations of the governing equation, boundary conditions, and initial conditions. In principle, this lets you solve partial differential equations without building a traditional numerical grid.
In practice, PINNs struggle with “stiff” equations, ones where the solution changes abruptly in some regions while remaining smooth elsewhere. These sharp gradients are common in physics: shock waves, boundary layers, chemical reactions with vastly different timescales. Concentrating training samples around the regions of rapid change seems like a natural fix, but it creates data imbalance that slows training and degrades accuracy. Newer techniques address this by adding preprocessing layers that remap the input distribution so the network sees a more uniform problem.18Computers & Mathematics with Applications. Stabilize physics-informed neural networks for stiff differential equations: Re-spacing layer These approaches regularize the solution gradients and have been shown to speed up convergence while improving accuracy on benchmark stiff problems.
Three-dimensional problems remain especially challenging for PINNs. When boundary conditions conflict at adjacent edges and corners of a domain, the solution can be discontinuous at those locations, and neural networks with smooth activation functions have a hard time representing discontinuities.19Archives of Computational Methods in Engineering. Stiff-PDEs and Physics-Informed Neural Networks The field is moving fast, but PINNs are not yet a universal replacement for classical numerical methods. They are best thought of as a complementary tool, particularly useful when traditional meshing is impractical or when you need a fast surrogate for an equation that must be solved thousands of times with slightly different parameters.
Analog Physical Solvers
Before the digital era, differential equations were routinely solved using analog computers, physical devices whose own behavior mirrors the equation you want to solve. That idea is making a comeback in a modern form. Researchers have built analog computing systems using ultrasonic metasurfaces, engineered surfaces that manipulate sound waves in precise ways, to solve both ordinary and partial differential equations. Simulations of these systems show high accuracy, with the physical wave propagation itself performing the computation.20Scientific Reports. Solving ordinary and partial differential equations using an analog computing system based on ultrasonic metasurfaces
A related approach uses light instead of sound. Photonic platforms built from epsilon-near-zero materials, which have unusual electromagnetic properties near certain frequencies, can solve partial differential equations in the analog domain. The appeal is speed and energy efficiency: photons moving through a carefully designed structure arrive at the answer at the speed of light, consuming far less energy than a digital processor crunching through millions of arithmetic operations.21Communications Physics. Approximate analog computing with metatronic circuits These systems are still in the proof-of-concept stage, and the equations they can handle are simpler than the full Navier-Stokes or Einstein field equations. But they represent a genuinely different approach to the problem, one where the physics of a device is harnessed to solve the physics of a question, sidestepping the bottleneck of digital computation entirely.
Why Some Equations Stay Hard
A pattern runs through these examples. An equation becomes hard when three things intersect: nonlinearity, coupling, and scale. Nonlinearity means that doubling the input does not double the output; instead, different parts of the solution feed back on each other. Coupling means that you cannot split the problem into independent pieces and solve them separately. And scale refers to the vast range of relevant lengths, times, or energies that must all be resolved simultaneously, from the tiny eddies in a turbulent flow to the large-scale currents, or from individual quantum states to the collective behavior of a material.
Interestingly, difficulty is not always permanent. Equations once thought intractable sometimes yield to new mathematical insight, as when the inverse scattering transform turned certain nonlinear wave equations into exactly solvable problems. New computational paradigms, from quantum algorithms that might tame six-dimensional plasma simulations to neural networks that learn solution shapes without explicit grids, keep shifting the boundary between “hard” and “feasible.” The equations themselves have not changed. Our tools for attacking them have, and that is where the story keeps moving.