NeuralFlowML Finite-Volume Graph and Boundary Encoding

Solver graph, not a generic point cloud. NeuralFlowML uses the finite-volume mesh topology itself. Nodes are control volumes, directed edges are cell-face incidences, and boundary edges preserve the physical boundary operator. The graph therefore encodes the same locality and orientation used by NeuralFlow flux assembly.

Directed cell-face graph

Every cell is a graph node. Every cell-face incidence is one directed edge. An interior face shared by cells and therefore contributes both and . A physical boundary edge has no destination cell and instead carries its boundary-family descriptor.

For an edge seen from owner cell , the compact geometry block is

with on an interior edge and on a boundary edge. Its dimension is .

Primitive normalization

The current fixed reference values are , and . Pressure and temperature are first kept in the admissible NeuralFlow range before logarithms are formed.

Finite-volume-inspired raw edge messages

For an owner state and a neighbor or differentiable boundary-surrogate state , define

The current raw edge-message families are

These quantities are features for learning the solver correction. They are not substitute conservation fluxes and are never used in place of the actual NeuralFlow residual.

Boundary-family buckets and geometric normalization

Raw messages are accumulated separately into interior, open-boundary, wall and symmetry buckets. Two local geometric scales are

Pressure/advection-like sums are normalized by ; thermal-gradient and velocity-diffusion sums are normalized by . Both raw and normalized forms are retained.

The operator-fraction vector contains local fractions of interior, far-field, pressure-outlet, wall and symmetry faces. The resulting node-input dimension is

giving in 2D and in 3D.

Versioned prescribed-boundary descriptor

The current boundary contract is schema version 2. It carries prescribed physical data, validity masks and family/subtype information. It is deliberately distinct from a snapshot of a solved boundary-face state.

with , , and . Therefore in 2D and 37 in 3D.

Auxiliary physical values

The auxiliary fields expose boundary physics needed to construct a differentiable local surrogate, including thermodynamic and kinematic information whose interpretation depends on the boundary family. Masks distinguish a meaningful prescribed value from an unused slot.

Boundary semantics in the learned solver

Far-field and pressure-outlet edges communicate the prescribed open-boundary state and pressure/backflow information. Wall edges communicate wall velocity and wall-type/thermal metadata. Symmetry edges communicate the symmetry operator rather than an arbitrary ghost-cell value. This preserves the distinction between different physical boundary operators inside the learned update.

Differentiable local boundary surrogate

The feature path constructs a local differentiable surrogate state from the owner primitive state and prescribed boundary data. That surrogate is used only to form graph features; NeuralFlow still applies the actual production boundary operator when assembling for the physics objective.

Learned hidden message passing

After encoding the physical node vector, NeuralFlowML applies a shared recurrent graph stage. In schematic form,

where the same parameter set is reused across repeated graph stages. Degree normalization prevents the latent update magnitude from growing simply because a cell has more incident graph edges.

What the graph is designed to learn

The graph supplies local state, geometry, boundary semantics and finite-volume-inspired directional signals. The neural operator is then trained to infer a correction that reduces the actual NeuralFlow finite-volume imbalance. In that sense the graph is an encoding of the nonlinear-solver neighborhood, not a dataset representation of a completed CFD solution.