Skip to content

PDEs and boundary conditions

Keep three choices separate: where points are located, what PDE they must satisfy, and what conditions apply on the boundary. A PDE preset does not choose a reference solution or boundary conditions for you.

Choose a preset or an expression

Use Predefined PDE for a tested family, or PDE residual for a restricted mathematical expression. Both fill the same PDE input on the trainer: choose one producer, not two residual blocks connected sequentially.

The Helmholtz preset exposes squared wavenumber and its forcing expression

The Helmholtz preset exposes squared wavenumber and its forcing expression. Open full size.

The current families include Poisson, reaction–diffusion, 1D advection–diffusion, Helmholtz and steady 2D incompressible flow. Coefficient meanings and signs differ; read the equation displayed in Properties. The example catalogue shows complete configurations. Loading an example is the simplest way to start.

Custom expressions use an allowlisted mathematical language, not Python scripts. For example, laplace(u)+c is a scalar Poisson residual. D(u,x) denotes a derivative. See the expression and component reference before writing a different operator; unsupported syntax is rejected rather than executed.

Assign each named boundary

Select Boundary condition, choose a side, and assign its supported type and target. Dirichlet specifies a field value; Neumann specifies its outward normal derivative. A custom operator can express supported Robin-like conditions.

Boundary-condition settings assign physics to the independently sampled sides

Boundary-condition settings assign physics to the independently sampled sides. Open full size.

Normal orientation matters. The scalar Neumann value is an outward derivative, not automatically a conductivity-scaled flux. Current scalar problems require a Dirichlet anchor. Adjacent Dirichlet values must agree at shared rectangle corners. The dedicated hard-Dirichlet method has stricter equal-constant box conditions.

For steady flow, boundary settings apply to u, v and p, and a pressure gauge fixes the pressure reference. Start from 2D incompressible flow for a complete working graph; scalar and vector contracts are not interchangeable.

Reference values are optional

A Reference block is needed only when you request a Comparison. Examples may supply an explicit expression; your own problem may have no exact solution. Do not add a reference merely to make the graph look complete.

Next: Choose networks and PINN methods.

For physical time dependence, start with Burgers — 1D + time. It uses (x,t) coordinates, two spatial BC traces and a separate initial profile, with no final-time condition. Its preset uses u_t + u*u_x - nu*u_xx = 0.

Arrows choose where the PDE is enforced

The PDE block accepts Collocation points and Additional boundary points (optional). This applies to both PDE residual and Predefined PDE.

  • An interior source connected to Collocation points enforces the PDE there.
  • A boundary source connected to the additional input adds those boundary rows to the PDE loss. Without that arrow, boundary rows are not added.
  • A boundary source can instead connect directly to Collocation points for a boundary-only residual; remove the unused interior sampler in that graph.
  • The independent arrow from Boundary data/points to Boundary condition controls BC losses. Connecting a boundary source to the PDE does not remove its BC loss.

Interior and boundary arrows feed the PDE independently

Both arrows: residual evaluated on the combined point rows; boundary operators are still evaluated separately through the Boundary condition block.

Without the extra boundary arrow the PDE uses only interior points

No boundary arrow to the PDE means no extra boundary PDE rows.

New templates and File → Examples include the boundary-to-PDE arrow. Existing saved models retain their original wiring and sampling behaviour. The low-level PoissonConfig API keeps pde_points='interior' for compatibility; scripts can explicitly select 'both' or 'boundary'. Example scripts select their intended mode explicitly. Points are concatenated without deduplication: shared corners or repeated input rows retain their sampling multiplicity. The PDE term is one mean squared residual over the selected rows, not an additional BC term. Preset forcing is evaluated at the added rows too. For transient problems, the connected boundary source may also contain initial points. Periodic examples include paired faces; pressure gauges and mean-velocity anchors are not geometric boundary point sets.

RAR retains the boundary PDE anchors while growing the interior set. Importance sampling resamples interior rows with inverse-probability weights and retains boundary anchors at unit weight. RAR's maximum is the total active PDE-row budget, including connected boundary anchors. Fixed SA/RBA masks include all selected PDE rows. Expert causal weighting bins added rows by their physical time, preserving complete coordinate tuples, and Scale refreshes the selected point sets together.

Integral loss is different: its PDE term is a domain-volume quadrature. Disconnect the extra boundary PDE arrow for that mode; otherwise validation explains the incompatibility. Boundary operators keep their own boundary measure. Margam never assigns volume quadrature weights to extra boundary points automatically.

Boundary collocation is an explicit modelling choice. Imported points are not geometrically filtered, and users may omit nonsmooth corners or incompatible initial traces from the PDE dataset. There is no universal paper convention: for example, the upstream gPINN Burgers residual-only setup explicitly uses zero boundary/initial samples. The graph supports either choice.

The CPU and CUDA checks verify selected PDE row counts and exact same-device continuation for baseline, importance, gPINN, RBA and Expert training. These are correctness checks, not accuracy benchmarks.

CUDA_VISIBLE_DEVICES='' .venv/bin/python -m examples.pde_collocation --device cpu --output build/pde-collocation-cpu.json
.venv/bin/python -m examples.pde_collocation --device cuda --output build/pde-collocation-cuda.json

Set individual boundary loss coefficients

Each named constraint in Boundary condition has a loss coefficient, default 1. For ordinary scalar problems the objective is

PDE_weight × PDE_loss + BC_weight / number_of_constraints × Σ(coefficient_i × BC_loss_i).

For two boundaries, overall BC weight 3, and coefficients 2 and 0.5, their fixed multipliers are 3 and 0.75. Increasing one does not renormalize the others. The raw errors and point counts are unchanged. Use a positive coefficient between 1e-6 and 1e6; zero is not a way to remove a required anchor.

Transient initial and periodic constraints, and coupled flow component constraints, also have individual coefficients under their actual constraint names. Pressure gauges retain their existing separate gauge weight, multiplied by their own coefficient. Paper presets retain their previous normalization. SA, LRA and Expert can apply additional adaptive weights; a manual coefficient does not freeze or replace their balancing algorithms. In integral mode the unweighted term already includes its physical measure.

The Training Monitor can show raw and actual weighted loss for a selected constraint. The Jorvik example shows the controls with independently named curved boundaries.

The Python setting is PoissonConfig(bc_weights={'inner': 2.0, 'outer': 0.5}). Keys are constraint names (initial for a time-zero set, left:u for a coupled velocity component). Inactive entries do not contribute to the current objective. Model format 27 stores the mapping; older models default to unit coefficients. Changing an active coefficient is rejected when loading a checkpoint from a previously different objective.