The convenience yield, the implied benefit of holding physical commodities, is a latent state inferred only from the term structure of futures prices. Traditional methods include the Kalman Filter (KF), which assumes linear-Gaussian dynamics and never verifies the recovered states against the governing partial differential equation (PDE). We introduce the Affine-Partitioned, Physics-Enforced Network (AP-PEN), which inverts the two-factor Gibson-Schwartz model for the instantaneous convenience yield of West Texas Intermediate (WTI) crude oil futures. Rather than penalising the system for breaching the governing PDE, AP-PEN imposes the model's affine coefficients analytically, so a single network need only parameterise the latent convenience yield path. The resulting architecture satisfies the PDE by design, leaving no residual loss to learn. Physics instead enters through a transition likelihood on the recovered path under the physical measure, alongside no-arbitrage and economic inequality constraints balanced by a gradient-norm scheme adapted from Hoshisashi et al.'s Whack-a-mole Online Learning (WamOL). Validated on simulated and real WTI data (2015-2026), AP-PEN's no-arbitrage variant uniquely avoids collapsing into implausible estimates during the April 2020 market dislocation, tracking the KF three times closer than a naive constant-average baseline, but at the cost of degenerate structural parameters. On clean synthetic data, these constraints prove counterproductive, with the closed-form least squares estimate outperforming every network variant. This work extends WamOL and derivative-constrained physics-informed neural networks to commodities, contributing an identifiability analysis and a hard-constrained estimator rather than a residual-based architecture.