No solution dataset
Training uses no example solutions, converged CFD snapshots, labeled state fields or prescribed solver trajectories.
NeuralFlow Graph Neural Operator CFD solver
A learned solver trained directly from CFD numerics
NeuralFlow is LIKUA's Graph Neural Operator (GNO) CFD solver, trained without example solutions, reference flow fields or supervised datasets. The GNO learns an iterative solution operator on the unstructured computational graph, with its training signal generated directly by the finite-volume residual, numerical fluxes, discrete Jacobians and boundary-condition treatment.
Zero example solutionsThe numerical method is the teacher. The Graph Neural Operator is trained to solve the discrete equations, not to imitate precomputed CFD results.
No reference solutionsNo labeled flow fields, converged examples or teacher trajectories are required during training.
Finite-volume conservationThe learning signal is formed from the same discrete balance equations used by the solver.
Solver numerics as supervisionNumerical fluxes, Jacobians and boundary treatment define the training direction.
Graph Neural Operator updateThe GNO performs iterative flow-state updates on the unstructured computational graph.
Training principle
The numerical method is the teacher
NeuralFlow is not trained to match CMPS solutions. Its Graph Neural Operator is trained to solve the same discrete conservation system using numerical feedback generated by the solver itself.
Solver-native objective
Mass, momentum and energy feedback is produced by finite-volume flux imbalance, source terms and boundary-condition numerics.
A Graph Neural Operator solver
The GNO generates successive flow-state corrections and learns from the numerical quality of its own multi-rollout trajectory.
Graph Neural Operator solver loop
The GNO is trained as the solver iteration
At each rollout, the Graph Neural Operator maps the current flow state on the unstructured computational graph to the next state correction. The finite-volume numerics evaluate the resulting state and generate the information required to train the next solver update.
Learning to solve, not learning a solution
Training begins from an admissible initial state, not from a known answer. The GNO generates successive updates to the primitive variables and therefore constructs its own path toward a converged solution.
During training, the CFD numerics evaluate conservation imbalance, numerical fluxes, sources, boundary conditions and discrete sensitivities. This solver-generated feedback trains the Graph Neural Operator without any reference flow field.
Initialize a valid flow stateGeometry, mesh, initial conditions and boundary conditions define the problem; no target solution is supplied.
Advance the state with the GNOThe Graph Neural Operator produces the next update to velocity, pressure and temperature on the computational graph.
Evaluate the discrete conservation systemThe finite-volume numerics calculate flux imbalance, source contributions and boundary-condition effects.
Differentiate the solver numericsSolver-computed discrete Jacobians and automatic differentiation provide physical sensitivities for training.
Train through the generated rolloutThe GNO learns from its own sequence of states rather than from an example solution or teacher trajectory.
Solver-native training: The terminal rollout state is evaluated through the scaled conservation objective. Solver-computed flux Jacobians and automatic differentiation propagate this physical training signal back through the multi-rollout Graph Neural Operator.
Not supervised learning and not a classical PINN: NeuralFlow uses no example solution or target field. It learns from the discrete finite-volume numerics themselves, including mesh connectivity, numerical fluxes, source terms, boundary treatment and the linearized structure of the solver.
Self-generated training trajectory
The solver generates its own training signal
Every state encountered during training is produced by the Graph Neural Operator's own rollout and evaluated by the CFD numerics. No converged example field is supplied as the answer.
Research targets
Focused first on difficult conservative flow problems
The development program starts from regimes where solution surrogates and pointwise physics losses are least convincing: compressible flow, strong gradients, boundary-sensitive solutions and complex unstructured meshes. NeuralFlow instead trains a Graph Neural Operator directly through the discrete finite-volume solver.
Compressible internal and external flow
Research on Graph Neural Operator solution of steady finite-volume systems across low-speed and shock-containing compressible regimes.
Propulsion and high-speed aerodynamics
Long-term targets include nozzles, intakes, combustors, high-speed vehicles and other tightly coupled flow systems.
Boundary-layer-sensitive configurations
Training studies explicitly examine how wall and characteristic boundary information propagates through the learned solver graph.
Unstructured mesh generalization
The Graph Neural Operator acts on mesh connectivity and geometric features rather than a fixed Cartesian image grid.
Transient and turbulent extension
Time-accurate, turbulent and scale-resolving applications remain research-roadmap items requiring dedicated validation.
Multiphysics extension
Future work will examine learned solution of additional transport equations and strongly coupled CMPS physics.
Development status
Clear separation between implemented research and future capability
NeuralFlow remains an active R&D program. Public claims on speed, robustness and generalization will be tied to controlled benchmark evidence. Training without reference data does not remove the need for independent verification against accepted CFD benchmarks and experiments.
Implemented foundations
- Training without example or reference solutions
- No supervised CFD field dataset or teacher trajectory
- Graph Neural Operator solver updates of primitive flow variables
- Finite-volume residuals, fluxes, sources and boundary numerics as the training signal
- Solver-computed discrete flux Jacobians
- Forward-mode automatic differentiation inside the CFD numerics
- Reverse-mode training of the Graph Neural Operator
- Terminal physics objective after the generated multi-rollout trajectory
- In-solver C++ training and execution pathway
Work still required
- Benchmark-based speed, accuracy and robustness qualification
- Generalization across geometries, meshes and operating conditions
- Stable training over wider initial-condition ranges
- Large three-dimensional model and memory scaling
- Turbulent, transient and multiphysics validation
- Production workflow integration through GUIX
- Clear failure detection and fallback to the validated solver
LIKUA simulation stack
Validated physics, a Graph Neural Operator solver and operational use
NeuralFlow is the Graph Neural Operator learned-solver research layer of LIKUA's engineering platform. Its parameters are trained from solver numerics rather than from example CFD solutions.
Research and technical collaboration
Develop the next generation of Graph Neural Operator CFD solvers
LIKUA is open to technically grounded collaboration on benchmark design, solver-native learning, Graph Neural Operator solver development and validation for demanding aerospace and industrial flow problems.
Contact LIKUA