Procedural Generation of Game Maps as a Constraint Satisfaction Problem: An Euler-based Conditional Flow Matching Approach

Abstract

This work investigates the application of Euler-based Conditional Flow Matching (CFM) to the procedural generation of tile-based 2D game maps, framed as a Constraint Satisfaction Problem (CSP) in which generated maps must satisfy implicit local and global structural rules learned from a training corpus. A generative pipeline is implemented from first principles in C# with TorchSharp: a two-level U-Net parameterizes a time-dependent vector field trained via the Conditional Flow Matching objective, and inference proceeds through an Euler ordinary differential equation solver at fifty function evaluations per sample. The Video Game Level Corpus Super Mario Bros distribution provides fifteen levels across three biomes (Overworld, Underground, Treetop) as the training set. Development proceeded across four iterations, each targeting the dominant failure mode observed in the previous iteration's output as measured by a six-category failure-mode analyzer. Iteration 1 established the baseline pipeline; Iteration 2 explored class-balanced loss variants and biome conditioning; Iteration 3 tested capacity-and-duration scaling and produced a negative result identifying a fundamental quality ceiling of the CFM plus argmax decoding formulation; Iteration 4 introduced a deterministic post-generation repair pipeline of eleven rule-based primitives. Across thirty generated chunks, the hybrid pipeline reduced three hundred and nine unintended structural violations to zero, with sixty-five intentional Treetop ground gaps preserved as a biome feature.