physics informed neural network

In a rubber band stretched in the lab, three different elasticity mechanisms operate simultaneously. At small strains, Hooke's law prevails; at intermediate strains, entropic chain elasticity; at large strains, strain-induced crystallization.

A piston in a cylinder with gas usually moves when the gas pushes on it. But there is a piston that moves when the space in front of it is empty. Every five seconds, it checks the space near its wall, and if no one is there, it takes a step inward, compressing the gas without direct mechanical force.

In the Antarctic ice, a ghost particle leaves behind only a faint flash of blue light. The IceCube detector captures these flashes, and from them we need to infer where the particle came from. The PolarBERT model has learned to do this, but what exactly it sees inside has remained a mystery.

Diamond is not just a gemstone. Hidden inside its crystal lattice are defects that sense magnetic fields better than many instruments. One such defect, the NV center, acts as a tiny compass, but its needle wavers from heat and noise. For the compass to point accurately, calibration is needed.

Strike a bell, and it rings at its own pitch. A black hole, after being disturbed, also rings — except instead of sound, it's the fabric of space itself that shivers. The frequencies of this ring can tell you a lot about the hole: how fast it spins, what its mass is.

If you pour honey into a colander with a hundred small holes, it will drip in thin streams, but each stream will swirl around the edge of the hole. Engineers need to understand such swirls: they affect how quickly water filters, how oil comes out of a reservoir, or how a drug seeps through tissues.

One hundred milliseconds. In that time, after a lightning strike or a short circuit, the power system either returns to normal operation or loses stability. A new-generation Neural Network Oscillations learns to sense this moment, like an experienced tightrope walker anticipating a gust of wind.

A neuron receives thousands of signals on its branching processes — the dendrites. Some signals push the cell toward firing (excitation), others hold it back (inhibition). For a long time, it was thought that the neuron simply adds up all the 'fors' and 'againsts.' But in real dendrites, inhibition works differently: it can not subtract but divide the excitatory signal.

Many natural processes are described by differential equations, but scientists often have only fragmentary data — a sort of torn map. The new computational method works like a cartographer who reconstructs a complete map of the terrain from a few landmarks.

A typical tracking program resembles a navigator steering a ship by the stars. But as soon as the sky clouds over, coordinates drift, and entropy (a measure of uncertainty) builds up in the data. The new algorithm gets a ‘co‑pilot’ — a module that monitors how confident the main navigator is.

When an unmanned aerial vehicle learns from an experienced pilot, it memorizes not individual maneuvers but entire safe corridors — the trajectories along which the human mentor moved. This approach is called imitation learning: the machine mimics expert demonstrations.

A person in a boat turns by shifting their body weight. An unmanned vessel uses the same trick: a sliding weight inside the hull does the steering. Its movement is controlled by a neural network, anticipating the chaos of waves and gusts of wind. The propeller pushes the boat forward, while the weight shifts balance — like moving from one foot to the other.

Neutron stars are the last clumps of visible matter before collapsing into a black hole. In their interiors, matter is squeezed to trillion-fold densities. Here, nuclear forces and gravity weave into an exotic substance inaccessible to any earthly particle accelerator.

Neurochips work differently from conventional computers, which are built according to the blueprints of von Neumann with separate memory and processor. At the core of these chips is a network of simple cells, similar to neurons, and, like water that evenly fills containers of any shape, they easily find a solution without unnecessary steps.

The railway overhead contact line resembles a galaxy: energy flows through it like gravity, with trains becoming planets. The more trains draw current simultaneously, the stronger the voltage fluctuations — gravitational waves ripple through the system, threatening blackouts.

The float in a toilet tank is an ideal controller: sensing the water level, it instantly brings the system into equilibrium without looking back at the past. Engineers applied this principle to neural networks.

Machine learning has long ceased to be just a tool — it is becoming an investigator, independently fishing out the laws of nature from data. But how does a silicon mind come to understand?

The number −0.000459 is no random guest in the Blasius equation. It's the coefficient of η⁵, defining the boundary layer on a flat plate — a thin whisper of air friction against metal. Standard numerical methods like Runge–Kutta yield only an approximate curve. Physics-informed neural networks (PINNs) try to guess the entire solution. Often — unsuccessfully.

Ordinary neural networks learn from examples, like a person memorizing tunes by ear. But networks with physical laws embedded from the start resemble a musician who knows notes and harmony: they can play a new piece from just a few chords. Instead of mindless memorization, they use equations like energy conservation or motion.

A handful of sand falls onto a table: each grain rolls down until it rests in a hollow. That's how a potential works: the table's surface tells the grains where to go. But in a river with whirlpools, there's no single basin everything flows into. Water swirls, returns, and meanders. For such systems, the usual map of 'pits and hills' doesn't apply.

Population growth from a distance looks like a train picking up speed: it crawls at first, then gains momentum. But look closely at the data, and the picture blurs. One measurement shows a dozen individuals, another a hundred. Classical models wipe away this scatter like annoying smudges on a lens. They assume the noise is uniform and simply ignore it.

A chef doesn't waste ingredients cooking the whole dish to check the taste: they just taste a drop of sauce. The same logic underlies the "information matching" method for predicting metal strength. Instead of expensive calculations of every atom's behavior, scientists "taste" only simple indicators linked to plasticity — a metal's ability to bend without breaking.

A boiling kettle with the lid on doesn’t show the bubbling, but the sound and vibration betray the intensity of the water boiling.

A laser beam strikes bone, and it responds with a nearly inaudible ultrasonic echo. Hidden in that sound is information about every microcrack and every fluid-filled pore. The trouble is that conventional computer programs don't understand this language: they see the signal but don't know the physics that produced it.

The vessels inside the skull resemble a mountain road's hairpin turns — sharp bends, narrowings, unexpected curves. When a traffic jam forms on such a serpentine, blood stops reaching the brain properly. To predict trouble, doctors need to see the entire flow: where it accelerates, where it swirls into eddies. The full picture comes from a special MRI, like slow-motion cinematography of fluid.

A thermostat keeps a room at 20 degrees. Open a window in winter, and it kicks on the heat to restore equilibrium. The brain does something similar: it constantly predicts what will happen a moment ahead and corrects itself when wrong. That's how it learns.

A neutron spectrometer is like a blind geologist who taps on rock and guesses what's inside by the sound. It sends a beam of neutrons into a crystal and watches how they scatter. From this pattern, you can deduce how atomic magnets, spins, oscillate. But where exactly to dig and how long to listen used to be decided by a human. Now, the TAS-AI algorithm has taken the helm.

A neural network trained on collisions of protons and electrons transferred its skills to a completely different experiment — catching neutrinos. Like a chef who learned French cuisine and then, without preparation, baked an excellent Uzbek plov.

Magnetic monopoles are like halves of a magnet that cannot exist alone in ordinary materials. But in spin ice, they appear in pairs connected by an invisible thread — a Dirac string. Previously, such monopoles were only seen in artificial systems at room temperature, but their quantum motion remained a mystery.

Water in a river flows around rocks, air bends around an airplane wing: these motions are described by equations that are hard to solve even on a powerful computer. Neural networks can approximate such solutions, but they often err when data is collected at only a few points or is noisy.

Astronomers measure distances in the universe using explosions of special stars — Type Ia supernovae. These flashes always have nearly the same brightness, like light bulbs of known wattage. Knowing how much light reached us, you can calculate how far away the explosion is.

When a typical computer searches for a law of nature from data, it tries to guess a complete formula in one go. It's like a cook who, without knowing the recipe, dumps all the ingredients into a pot at once and hopes for soup. Most often, the result is an inedible mess: the formula is long but meaningless, and it fails on new data.

A chunk of nuclear fuel inside a reactor is not just a solid. When heated, the uranium atoms in it start to 'wiggle' their magnetic moments, and this motion changes thermal conductivity and other properties.

A material is like a person: if you stretch it for a long time, it remembers and behaves differently. Engineers have spent years fitting formulas to describe such memory. Now a neural network does it on its own. The new architecture, iCKAN, takes test data and converts it into equations.

Sunspots are traces of magnetic fields that emerge on the surface. They live their own lives: spots closer to the equator tilt more, while those farther away tilt less. These two effects, like two knobs on an old radio, control how much magnetic field reaches the Sun's poles and accumulates there for the next cycle.

Stars on the outskirts of galaxies rotate faster than predicted by the laws of physics if only visible matter is considered. This paradox was uncovered in the 20th century by Vera Rubin and Fritz Zwicky.

Earth's magnetic field is like an invisible river that meanders beneath our feet. Navigators and geologists rely on maps of this river, but instruments only take readings at scattered points, leaving blank spots in between. Filling those gaps is like trying to sketch the river's course from a few pebbles found on the bank.

Inside a battery, like a layered cake, different solid layers sit side by side. The meeting point, the interface, behaves capriciously: stresses concentrate there, and over time the battery ages. To understand these processes, we need computer models that see every atom. But accurate calculations are too expensive, and fast ones are too crude.

Inside a proton, gluons buzz around like bees in a hive, gluing quarks to each other and to themselves. The higher the collision energy, the more gluons emerge from the fog. Physicists have long sought a way to describe this swarm with a single simple function, but each time, they had to tweak the form to fit a specific experiment.

In the Large Hadron Collider's detector, each proton collision leaves thousands of points — tracks of flying particles. A neural network learned to spot patterns in this scatter and predict what happened at the moment of impact.