The dense PINN is the number a trader should keep in mind. Trained exactly like BeTINN, it gets a delta MAE of 0.0188 with 3,631 parameters. BeTINN gets 0.0180 with 3,595.
What BeTINN changes
The authors want one network to price a zero-dividend European call throughout a bounded box of Heston parameters. It takes time to maturity, log-moneyness x = log(S/K), instantaneous variance v and the five model parameters r, ρ, κ, θ, σ. Those are eight inputs. Training uses no prices: the loss combines the Heston PDE residual, the payoff at τ = 0 and two log-moneyness boundary conditions. Optimization uses 64-bit Levenberg-Marquardt, with a 3,000-step cap and early stopping on held-out physics loss. Greeks come from autodiff. A semi-analytic Heston oracle supplies the reference values on frozen 10,000-point panels.
BeTINN builds on TINN, whose weight blocks each have a base tensor and one learned direction scaled by time. BeTINN makes that scale depend on the five Heston parameters too. Each block can move along q directions (q = 4 in the final model), and a small residual network reads all eight inputs directly. Baseline parameter counts fall within 2.4% of BeTINN's. The authors selected q and the widths using six development seeds and a development panel. They then froze the design for six fresh seeds and a separate panel. That separation deserves credit.
On the confirmatory panel, BeTINN has the lowest mean MAE for price, delta, strike-scaled gamma (KΓ) and variance sensitivity u_v. Its price MAE is 0.00746, beside 0.00857 for the PINN, 0.00910 for TINN and 0.00974 for a physics-informed DeepONet.
How much of the lead survives a close look?
The price advantage over the PINN is about 13%; for u_v it is about 11% (0.00712 vs 0.00800). Delta and KΓ gains are roughly 4% (0.143 vs 0.149 for KΓ). The authors call those gains "more modest" and the PINN comparison "less uniform for the derivative metrics". Their case rests on BeTINN's lowest mean MAE across all four metrics jointly and its wins on 4 to 5 of 6 paired seeds. Paired by seed, it beats the PINN 5/6 times on price, 4/6 on delta, 4/6 on KΓ and 5/6 on u_v.
The delta lead is thin. Its 0.0008 gap falls inside BeTINN's own delta standard deviation of 0.0010. On price, the seed standard deviation is 0.00097, close to the 0.00111 gap. The PINN even has the lower mean KΓ RMSE: 0.385±0.010 versus 0.397±0.022.
The change between panels matters more. BeTINN's price MAE rose from 0.00625 in development to 0.00746 in confirmation. The PINN moved from 0.00859 to 0.00857. TINN rose from 0.00759 to 0.00910, which suggests a harder panel accounts for part of the change. BeTINN's lead over the PINN still shrank from 0.00234 to 0.00111. Winner selection likely accounts for some of that loss. The q sweep was non-monotone: q = 4 yielded 0.00625, q = 5 yielded 0.00736 and q = 6 yielded 0.00696. Choosing the best of six q values and four backbone widths on the same six development seeds invites winner's bias. The confirmatory run gives it room to show.
Would the delta gap pay for itself?
An end-of-day desk could calibrate Heston to a name's surface, ask the surrogate for delta at each strike and rebalance. A delta MAE of 0.018 means an average error of 1.8 shares per 100-share contract. BeTINN's 0.0008 advantage over the PINN amounts to 0.08 shares per contract, below lot rounding.
The paper cites hedging and risk analysis as reasons to care about differential accuracy. We did not find a hedging P&L or transaction-cost test. All points are synthetic Heston; ρ is restricted to [-0.9, 0.1], and there are no market quotes.
Speed against the reference matters as well. In the paper's convergence check, maximum differences for the semi-analytic oracle stay below 1.9e-5 across the full panel. The four surrogates' price MAEs span 7.5e-3 to 9.7e-3. A surrogate earns its place by outrunning that oracle. The timing table covers the four networks; we did not find an oracle timing.
Near expiry, the global gamma figure slips
Look at the 8-point subset with |x| ≤ 0.10 and τ ≤ 0.10, at the money and near expiry. BeTINN's KΓ MAE reaches 0.536. The PINN records 0.376, TINN 0.402 and PI-DeepONet 0.476. BeTINN ranks last there despite its global KΓ MAE of 0.143. Eight points cannot settle a ranking, and the global MAE offers little guidance for this corner.
The authors acknowledge that the PINN has "slightly lower KΓ error in several near-maturity subsets". They call the ATM near-maturity intersection "particularly demanding for curvature". "Slightly" describes the 204-point near-maturity subset: BeTINN gets 0.0753 and the PINN 0.0751. In the 8-point ATM subset, the comparison is 0.536 against 0.376.
The 48-point stress subset (x ≥ 1.6, τ ≥ 3.75, θ ≥ 0.7525) tells a mixed story. BeTINN has the worst price MAE, 0.0462, against TINN's 0.0385. It also has the lowest KΓ MAE, 0.000527 against 0.000639 to 0.000873 for the others. PI-DeepONet leads on delta, while the PINN leads on u_v. The authors' summary of this corner and the high-moneyness subsets is fair: "These results show that no single architecture uniformly dominates every localized regime and observable."
Finite-domain boundary approximations may contribute to errors in that corner, as the authors note. At x_max = 2, the paper's Δ = 1 approximation misses by 5.0e-3 on average, increasing to 1.2e-2 for τ ≥ 3.75 and θ ≥ 0.7525. Those are delta units, below every model's global delta MAE of 0.0180 to 0.0232. Our reading, and only ours, is that the corner ranking partly reflects each model's fit to an approximate constraint at x_max. At x = -2, the opposite boundary, the price mismatch averages 9.9e-3 and rises to 4.4e-2 in the same long-maturity, high-θ regime. The authors also show that e^{-x} and e^{-2x} coordinate factors produce the large low-moneyness delta and gamma errors. There, u_x and u_xx - u_x are no worse than they are globally.
Training time and transfer
BeTINN takes 432 s to train; the PINN takes 226 s. Single-point latency is 0.169 ms versus 0.101 ms, and throughput on a 10,000-point batch is 4.26e7 versus 7.27e7 samples per second. Either is quick enough for an end-of-day book. Transfer is the sharper concern.
On parametric Burgers, BeTINN's relative L2 is 0.000379 against TINN's 0.003053. A clear win. On parametric Wave, BeTINN is worst at 0.004134, behind the PINN at 0.002724 and TINN at 0.002101. The authors say it "does not retain the advantage observed on Heston and Burgers". Reducing q to 2 brings relative L2 to 0.00402, still behind both baselines. That leaves q as a per-problem choice. At a 1e-4 tolerance, every learned model has nonzero violation rates on the paper's six sign and shape checks; the oracle has none. The authors measure those properties without enforcing them during training.
The architecture is a clean idea, and the protocol is careful. A simulated delta-hedge on Heston paths with proportional costs would change my view if the 0.0008 delta gap appeared in P&L dispersion.