PINN CLI

PINN residual-bounding CLI.

This file implements a small interactive tool for bounding PDE residuals of a trained physics-informed neural network (PINN) over a box input region. Concretely, it:

• parses a compact PDE mini-language (see PdeAst / PdeParse),
• evaluates that expression using interval bounds for u, du, and d2u,
• computes those primitive bounds via IBP/CROWN-style propagation on a CROWN Graph,
• optionally tightens bounds by recursively splitting the input box (1D/2D).

This CLI is the interactive PINN residual-bound tool:

• use it when you want to inspect residual bounds interactively;
• use it when you are iterating on a PDE expression or input box;
• use the certificate checker when you want a stable artifact for docs, papers, or CI.

The stable artifact checker is: lake exe verify -- pinn-cert [NN/Examples/Verification/PINN/pinn_cert.json]

Run this CLI via the unified verification dispatcher: lake exe verify -- pinn-cli -- [flags] "<PDE>" x eps or for 2D: lake exe verify -- pinn-cli -- [flags] "<PDE>" x y eps

Examples:

• lake exe verify -- pinn-cli -- "u_xx + u" 0.0 0.1
• lake exe verify -- pinn-cli -- --backend=float "u_t - u_xx" 0.0 0.05

References:

• PINNs: https://arxiv.org/abs/1711.10561
• IBP: https://arxiv.org/abs/1810.12715
• CROWN: https://arxiv.org/abs/1811.00866

inductive NN.Verification.PINN.CLI.Backend : Type

Backend selection for the PINN CLI.

• float : Backend

def NN.Verification.PINN.CLI.parseBackendVal (s : String) : Option Backend

Parse the backend flag used by the PINN residual CLI.

def NN.Verification.PINN.CLI.parseFloat : String → Option Float

Parse a decimal Float literal used by CLI flags.

def NN.Verification.PINN.CLI.parseAlphaDecimal {α : Type} [Context α] (s : String) : Option α

Parse a non-scientific decimal string into a generic numeric α using Numbers and Context. Supports optional leading '-' and a single '.'.

def NN.Verification.PINN.CLI.parseAlphaDecimal.pow10 {α : Type} [Context α] (k : ℕ) : α

inductive NN.Verification.PINN.CLI.UBoundsMethod : Type

Select interval-only, forward CROWN, or backward CROWN bounds for the PINN output value.

• ibp : UBoundsMethod
• crownFwd : UBoundsMethod
• crownBwd : UBoundsMethod

def NN.Verification.PINN.CLI.computePrimsAt (g : MLTheory.CROWN.Graph) (ps : MLTheory.CROWN.Graph.ParamStore Float) (uMethod : UBoundsMethod := UBoundsMethod.ibp) (backend : Backend := Backend.float) : IO PdeAst.Prims

Compute primitives u, duX, duY, d2uX, d2uY at the unique output node (id=5) for 1D/2D models. Optionally, replace the u-interval with a tighter CROWN/DeepPoly bound.

def NN.Verification.PINN.CLI.loadWeightsOrDefault (weights? : Option String) (expectedInputDim : ℕ) (defaultGraph : MLTheory.CROWN.Graph) (defaultParams : MLTheory.CROWN.Graph.ParamStore Float) : IO (MLTheory.CROWN.Graph × MLTheory.CROWN.Graph.ParamStore Float)

Load optional PyTorch-exported PINN weights for the exploratory CLI.

Malformed files or mismatched input dimensions fall back to the provided built-in graph and parameters. That permissive behavior is intentional here: pinn-cli is an interactive inspection tool. Stable certificate checkers should reject bad artifacts instead.

inductive NN.Verification.PINN.CLI.Method : Type

User-facing bound method selected by --method.

• ibp : Method
• crownFwd : Method
• crownBwd : Method

@[implicit_reducible]

instance NN.Verification.PINN.CLI.instToStringMethod : ToString Method

def NN.Verification.PINN.CLI.parseMethodVal (s : String) : Option Method

Parse the --method value accepted by the PINN residual checker.

structure NN.Verification.PINN.CLI.Opts : Type

Parsed options for the PINN residual CLI.

• method : Method

  Bound propagation method used for the output interval.

• backend : Backend

  Runtime backend used for interval evaluation.

• weights? : Option String

  Optional PyTorch-exported PINN weights JSON.

• splitDepth : ℕ

  Recursive interval split depth for one-dimensional checks.

def NN.Verification.PINN.CLI.parseFlags (args : List String) : Except String (Opts × List String)

Parse recognized flags and return the remaining positional arguments.

def NN.Verification.PINN.CLI.main (args : List String) : IO Unit

Entry point for the PINN residual-bounding CLI.

This is an interactive tool registered as: lake exe verify -- pinn-cli -- ...

For certificate checking, use: lake exe verify -- pinn-cert [NN/Examples/Verification/PINN/pinn_cert.json]

Run: lake exe verify -- pinn-cli -- [--method=ibp|crown-fwd|crown-bwd] [--split-depth=N] [--backend=float] [--weights=PATH.json] "<PDE>" x eps or (2D): lake exe verify -- pinn-cli -- [flags] "<PDE>" x y eps

def NN.Verification.PINN.CLI.main.split1D (x eps : Float) (d : ℕ) (evalAt : Float → Float → IO (Float × Float)) : IO (Float × Float)

def NN.Verification.PINN.CLI.main.split2D (x y eps : Float) (d : ℕ) (evalAt : Float → Float → Float → IO (Float × Float)) : IO (Float × Float)