AI for Physics — PINNs, Operators & Simulation
This module surveys the machine-learning frontier for physical simulation: methods that either solve the governing differential equations of a physical system or replace an expensive numerical solver with a learned surrogate. The angle throughout is the AI/ML method, not the underlying physics derivations. For the broader programme see ai-for-science; for the molecular-sciences sibling see aifs-chemistry.
The field splits into two complementary families — physics-informed networks that embed the governing equations into the training loss, and neural operators that learn a reusable mapping across a whole family of problems — plus a fast-moving set of flagship large models (weather, climate, plasma, turbulence) that have moved from research demos to operational deployment between 2024 and 2026.
1. Physics-Informed Neural Networks (PINNs)
A PINN represents the solution of a partial differential equation (PDE) directly as a
neural network u(x, t; θ), then trains the network so that the PDE residual, boundary
conditions, and initial conditions are all driven to zero. The PDE itself enters the loss:
spatial and temporal derivatives are obtained by automatic differentiation of the network,
substituted into the governing equation, and the squared residual is minimized alongside
data-fit and boundary terms [1].
The defining property is that a PINN can be trained with zero labelled data — the physics supplies the supervision. This makes PINNs attractive for inverse problems, data-sparse regimes, and settings where a high-fidelity simulator is unavailable but the governing law is known [1]. Reviews from 2025–2026 frame PINNs as balancing physical consistency against data efficiency, while flagging persistent weaknesses in training convergence, multi-scale / high-frequency behaviour, and generalization [2].
A PINN solves a single instance of a problem. Change the boundary conditions, the forcing, or a coefficient, and the network must in general be retrained. Active research directions include domain decomposition, adaptive collocation-point sampling, frequency-enhanced (Fourier-feature) representations, and curriculum / hybrid differentiation schemes to stabilize training [2].
2. Neural Operators
Neural operators take the opposite stance: instead of fitting one solution, they learn the solution operator of a PDE family — a mapping between infinite-dimensional function spaces (e.g. from an initial condition or coefficient field to the solution field). Once trained, inference is a single forward pass, amortizing cost across many instances and giving mesh-independent, often resolution-transferable predictions [3].
- Fourier Neural Operator (FNO) — introduced by Li et al. (ICLR 2021), FNO parameterizes the integral kernel as a multiplication in the Fourier domain, giving a global receptive field and strong efficiency on regular domains with spectral content. It exhibits zero-shot super-resolution: train at one grid, evaluate at another [3].
- AFNO (Adaptive Fourier Neural Operator) — recasts FNO-style spectral mixing as an efficient token mixer inside a Vision-Transformer backbone, replacing quadratic self-attention with a global convolution in the Fourier domain. AFNO is the spatial-mixing engine behind FourCastNet [11].
- DeepONet — a branch/trunk architecture that learns operators by encoding the input function (branch) and the query location (trunk) separately. It is well suited to irregular geometries and sparsely sampled input functions, complementing FNO's strength on regular spectral domains [1].
PINNs and operators are complementary, not competing. A PINN delivers a high-accuracy solution for one configuration using the equation as supervision; an operator delivers fast approximate solutions across many configurations, typically needing simulation or observational data to train. Hybrids such as the Physics-Informed Neural Operator (PINO) add a PDE-residual loss to operator training, reducing the data requirement while keeping the amortized inference of an operator [1][3].
3. Software stack — NVIDIA PhysicsNeMo
NVIDIA PhysicsNeMo (the open-source successor to Modulus) is the most prominent end-to-end physics-ML framework. It spans the full spectrum from purely physics-driven PINNs to data-driven operators and generative models, and ships curated architectures including Fourier-feature networks, FNO, DeepONet, PINO, graph neural networks, point-cloud models, and diffusion models [3][4]. NVIDIA documents additional operator variants such as DoMINO (a decomposable multi-scale neural operator) aimed at generalization across engineering domains [3]. The platform is positioned for both research and industrial simulation acceleration (CFD, structural, electromagnetics) and was released as open source [4].
4. Weather & climate large models
Numerical weather prediction (NWP) integrates the atmospheric equations on a supercomputer. Data-driven models instead learn the next-state map from decades of ERA5 reanalysis (ECMWF's archive) and run inference in seconds to minutes, several orders of magnitude faster than full NWP [5][6].
- graphcast (Google DeepMind) — a graph-neural-network model producing deterministic medium-range global forecasts at 0.25° resolution, reported to match or exceed state-of-the-art NWP skill on many targets while running far faster [6].
- Pangu-Weather (Huawei) — a 3D Vision-Transformer model over multiple pressure levels, reported competitive with ECMWF's deterministic IFS on several medium-range metrics including tropical-cyclone tracks [5].
- FourCastNet (NVIDIA) — an AFNO/ViT model at 0.25° resolution, notable as an early demonstration that operator-style architectures scale to global high-resolution forecasting with modest memory [11].
- GenCast (Google DeepMind) — a diffusion-based ensemble model generating 15-day global ensembles at 0.25° resolution; the Nature paper reports it surpasses ECMWF's ENS on the large majority of evaluated targets, including extremes and tropical cyclones [5].
- Aurora (Microsoft) — a foundation model for the Earth system trained on diverse geophysical data and fine-tuned to tasks beyond weather (air quality, ocean waves, cyclones); published in Nature in 2025 under the pretrain–fine-tune paradigm [9].
- ECMWF AIFS — ECMWF's own Artificial Intelligence Forecasting System became operational on 25 February 2025, run alongside the physics-based IFS; the ensemble variant (AIFS-ENS, trained with a CRPS-based loss) followed into operations later in 2025 [7].
The 2025–2026 trajectory is toward operational AI forecasting and hybrid systems that combine learned components with physics-based integration, rather than wholesale replacement of NWP [7].
5. Other physics domains
- Fusion / plasma control — DeepMind and EPFL's Swiss Plasma Center used deep reinforcement learning to control the magnetic-coil actuators of the TCV tokamak, autonomously stabilizing a range of plasma configurations (elongated, negative triangularity, "snowflake", and simultaneous "droplets"); published in Nature in 2022 and a landmark for RL on a real high-frequency control system [8].
- Turbulence / CFD surrogates — learned subgrid closures and full-field surrogates replace or accelerate expensive solvers. Differentiable physics embeds an ML closure inside a differentiable solver so the model is trained end-to-end against the resolved dynamics, combining solver generalization with learned flexibility [10].
- Lattice QCD — ML accelerates Monte-Carlo lattice simulations via normalizing flows (improved sampling), contour deformations (sign problems), control variates (signal-to-noise), and surrogate observables [10].
- Cosmological emulators — surrogate models stand in for costly N-body / hydrodynamic simulations across parameter space, enabling rapid inference; part of the broader emulator-and-surrogate trend in ai-for-science.
- Differentiable simulation — making the simulator itself differentiable lets gradients flow through physical dynamics for control, design, and inverse problems, unifying several of the above threads [10].
6. Methods table
| Method | What it learns | Data need | Example |
|---|---|---|---|
| PINN | Single-instance PDE solution u(x,t) |
Zero labelled data possible (physics loss) | Forward/inverse PDE solve [1] |
| FNO | Solution operator (spectral kernel) | Simulation/observational data | Darcy flow, weather [3][11] |
| AFNO | Operator as ViT token mixer | Data | FourCastNet weather [11] |
| DeepONet | Operator via branch/trunk | Data; handles sparse inputs | Irregular-geometry operators [1] |
| PINO | Operator + PDE-residual loss | Reduced data (physics-regularized) | Hybrid operator solve [1][3] |
| GNN emulator | Next-state map on a mesh/graph | Reanalysis data | GraphCast [6] |
| Diffusion ensemble | Stochastic forecast distribution | Reanalysis data | GenCast [5] |
| Differentiable solver + ML | Closure / correction inside solver | Resolved-dynamics data | Turbulence closure [10] |
PINN vs. neural operator
| Axis | PINN | Neural operator |
|---|---|---|
| Output | One solution instance | A solution operator (family) |
| Supervision | Governing equation (can be data-free) | Typically data; PINO adds physics loss |
| Reuse | Retrain per new configuration | Amortized: one pass per instance |
| Inference speed | Slow (per-problem optimization) | Fast (single forward pass) |
| Strength | Accuracy, inverse problems, data-sparse | Throughput across many instances |
| Weakness | Generalization, training convergence | Out-of-distribution extrapolation |
7. Open problems
- Generalization & extrapolation. Operators interpolate well within their training distribution but degrade out-of-distribution — new boundary conditions, geometries, forcing regimes, or unseen physical parameters. PINNs face the dual problem of training convergence on stiff or multi-scale systems [2].
- Long-horizon stability. Autoregressive emulators (weather, fluids) can accumulate error and drift over long rollouts; maintaining physical conservation and avoiding blow-up over many steps remains an open challenge.
- Multi-scale & high-frequency content. Spectral bias makes sharp gradients, shocks, and fine-scale turbulence hard to capture; frequency-enhanced and decomposition methods are active responses [2].
- Verification & physical guarantees. Learned surrogates do not inherently respect conservation laws or hard constraints; quantifying uncertainty and guaranteeing physical admissibility is unresolved.
- Hybrid integration. The strongest near-term direction is coupling learned components with physics-based solvers (as in operational weather systems), but the design space for where and how to insert ML is still being mapped [7].
Sources
- https://arxiv.org/html/2511.04576v3 (2026-06-14) — survey: PINNs and neural operators for parametric PDEs
- https://www.mdpi.com/2076-3417/15/14/8092 (2026-06-14) — PINN methodological review
- https://docs.nvidia.com/physicsnemo/latest/user-guide/model_architectures.html (2026-06-14) — PhysicsNeMo architectures
- https://developer.nvidia.com/blog/physics-ml-platform-physicsnemo-is-now-open-source/ (2026-06-14) — PhysicsNeMo open source
- https://www.nature.com/articles/s41586-024-08252-9 (2026-06-14) — GenCast (Nature, probabilistic forecasting)
- https://deepmind.google/blog/gencast-predicts-weather-and-the-risks-of-extreme-conditions-with-sota-accuracy/ (2026-06-14) — GenCast / GraphCast context
- https://www.ecmwf.int/en/about/media-centre/news/2025/ecmwfs-ai-forecasts-become-operational (2026-06-14) — ECMWF AIFS operational
- https://www.nature.com/articles/s41586-021-04301-9 (2026-06-14) — DeepMind/EPFL tokamak magnetic control (Nature)
- https://www.microsoft.com/en-us/research/publication/aurora-a-foundation-model-for-the-earth-system/ (2026-06-14) — Aurora foundation model
- https://arxiv.org/html/2502.02670v1 (2026-06-14) — ML for lattice simulations / differentiable physics
- https://arxiv.org/pdf/2202.11214 (2026-06-14) — FourCastNet (AFNO)