A neural-network-inspired F-expansion approach for analytical solution families of nonlinear partial differential equations

This work presents a neural-network-inspired symbolic ansatz framework based on the classical Riccati/𝐹-expansion structure for constructing analytical solution families of nonlinear PDEs. The network terminology refers only to the organization of the trial function through affine encoders, Riccati activation nodes, and polynomial or vector-valued output maps; no data-driven training or optimizer-driven physics-informed learning is used. The unknown parameters are determined by direct substitution and symbolic coefficient matching. Structural properties of the ansatz are established, including exact embedding of the classical 𝐹-expansion representation, a weighted 𝐿2-residual interpretation of coefficient matching in the traveling-wave setting, and approximation properties in the transformed variable. The Korteweg–de Vries and Benjamin–Bona–Mahony examples recover single-phase Riccati branches, whereas the two-dimensional coupled viscous Burgers system yields nine exact vector-valued families with two independent active affine encoders. A rank argument shows that these Burgers families are not reducible to a single affine phase on their regular domains. Quantitative residual checks and a sensitivity study support the analytical verification. The demonstrated benefit is structural: a modular organization of multi-encoder ansatzes and reusable residual identities, rather than a claim of computational speedup over trained neural networks or other analytical methods.