Beyond Time Induction: Structured Conditional Neural Solvers for Parametric Stochastic-Volatility Pricing

Jose Miguel Lara Rangel; Mixtli Vazquez Mendoza

cs.LGOct 13, 2026

Beyond Time Induction: Structured Conditional Neural Solvers for Parametric Stochastic-Volatility Pricing

Authors: Jose Miguel Lara Rangel, Mixtli Vazquez Mendoza

Organizations: University College London, UK · Aicraft, Mexico

Abstract

Physics-informed neural networks provide a flexible framework for learning parametric PDE solution families without labeled data, yet preserving accurate derivatives across heterogeneous regimes remains challenging. Structured neural solvers offer compact representations, but restrictive conditional parameterizations can limit adaptation across the solution family. We introduce BeTINN, a structured conditional architecture extending time-induced modulation through multi-directional time-parameter conditioning and a joint residual pathway. We evaluate BeTINN on the Heston stochastic-volatility PDE, a demanding parametric problem with mixed derivatives and economically relevant first- and second-order sensitivities. BeTINN improves joint accuracy across normalized option value and key sensitivity metrics relative to our baselines. Controlled ablations characterize the roles of its architectural components. We further find that validation physics loss is informative but does not fully determine differential fidelity, and that accurate solution values can coexist with localized derivative errors. These results support structured conditional adaptation and motivate derivative-aware evaluation when sensitivities are downstream quantities of interest.

Figures & tables

Figure 1: Architecture Description. (a) TINN uses time-conditioned modulation to adapt backbone parameters over time. (b) BeTINN generalizes time-induced modulation into a structured time–parameter conditional parameterization. Joint conditioning moves each backbone parameter group along multiple learned adaptation directions, yielding a structured family of regime-dependent networks. In parallel, a joint residual pathway captures direct state–time–parameter interactions, and both paths combine to produce the surrogate solution.

Method	Params	Price MAE $\downarrow$	$\Delta$ MAE $\downarrow$	$K\Gamma$ MAE $\downarrow$	$u_{v}$ MAE $\downarrow$
TINN	3,663	$0.00910\pm 0.00062$	$0.0197\pm 0.0009$	$0.158\pm 0.006$	$0.00853\pm 0.00046$
PINN	3,631	$0.00857\pm 0.00054$	$0.0188\pm 0.0005$	$0.149\pm 0.003$	$0.00800\pm 0.00049$
PI-DeepONet	3,511	$0.00974\pm 0.00063$	$0.0232\pm 0.0007$	$0.199\pm 0.016$	$0.0109\pm 0.0005$
BeTINN	3,595	$\mathbf{0.00746\pm 0.00097}$	$\mathbf{0.0180\pm 0.0010}$	$\mathbf{0.143\pm 0.005}$	$\mathbf{0.00712\pm 0.00089}$

Table 1: Model comparison on the frozen Heston evaluation panel. Entries report mean

\pm

std over six seeds; lower is better. Baselines are capacity-matched to BeTINN within

2.4\%

in trainable parameter count while preserving their native architectural structure.

Figure 2: Joint option-value and derivative accuracy. Price MAE is plotted against the MAE of

\Delta

K\Gamma

, and variance sensitivity

u_{v}

. Highlighted markers denote the final benchmark models, while light-gray markers show additional controlled configurations. The two sets correspond to their respective frozen evaluation protocols. Lower-left indicates lower error in both option value and the corresponding derivative. BeTINN consistently occupies the lowest-error region across all three comparisons.

Figure 3: Solution and derivative profiles across representative Heston regimes. We sweep log-moneyness

x

over three fixed

(\tau,v,r,\rho,\kappa,\theta,\sigma)

settings: interior

(1.84,0.52,0.08,-0.38,1.92,0.39,0.55)

, near-maturity

(0.07,0.57,0.16,-0.48,1.61,0.40,0.75)

, and stress

(3.78,0.64,0.15,-0.47,1.89,0.99,0.60)

. While all models closely match the price profiles, sensitivity errors are more visible in high-curvature and low-moneyness regions. The near-maturity slice shows differences around the sharp

K\Gamma

peak. Across the frozen evaluation panel, low-moneyness growth in

\Delta

and

K\Gamma

errors is amplified by the coordinate factors

e^{-x}

and

e^{-2x}

; in the long-maturity stress, finite-domain boundary effects may also contribute near the computational boundary.

Figure 4: Validation physics loss and downstream predictive error. Best validation physics loss is compared with MAE for price,

\Delta

K\Gamma

, and variance sensitivity

u_{v}

. Highlighted markers denote the 24 final benchmark runs from 4 architectures and 6 confirmatory seeds. Spearman correlations, reported above each panel, are computed over these model–seed pairs. Light-gray markers show additional controlled variants for visual context and are excluded from the correlation calculation. Validation physics loss is positively associated with downstream error, but does not preserve the same ranking across solution and derivative quantities.

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Constraint	Fixed coordinate	Training points
PDE interior	—	$15{,}000$
Initial condition	$\tau=0$	$100{,}000$
Left boundary	$x=x_{\min}$	$100{,}000$
Right boundary	$x=x_{\max}$	$100{,}000$

Appendix

Table 2: Physics-informed training panels used in the controlled experiments.

Method	Hidden / branch geometry	$q$	Conditioning / residual	Params	Mismatch (%)
BeTINN	$(20,20)$	$4$	conditioner $(12,10)$ ; joint residual $16$	$3{,}595$	$0.00$
TINN	$(37,37)$	$1$	temporal conditioner $(10,10)$	$3{,}663$	$1.89$
PINN	$(55,55)$	—	—	$3{,}631$	$1.00$
PI-DeepONet	branch/trunk/latent $39/39/39$	—	symmetric $1{:}1{:}1$ geometry	$3{,}511$	$2.34$
JointResidual (TINN)	$(34,38)$	$1$	temporal conditioner $(7,9)$ ; joint residual $16$	$3{,}577$	$0.50$

Appendix

Table 3: Architecture and capacity specification for the controlled models. Parameter counts are verified from the instantiated parameter trees. Baselines preserve their native architectural structure while remaining close to the final BeTINN parameter count.

Subset	Definition	Points
ATM tight	$\|x\|\leq 0.05$	253
ATM	$\|x\|\leq 0.10$	501
ATM wide	$\|x\|\leq 0.25$	1297
Very near maturity	$\tau\leq 0.05$	109
Near maturity	$\tau\leq 0.10$	204
Short maturity	$\tau\leq 0.25$	520

Appendix

Table 4: Regional diagnostic subsets. Definitions and numbers of selected points on the frozen

10{,}000

-point evaluation panel.

Model	Params.	Train time (s)	Best val. step	$B=1$ latency (ms)	$B=1000$ throughput	$B=10000$ throughput	Peak alloc. (MiB)
BeTINN	3595	$432\pm 26$	$72\pm 21$	$0.169\pm 0.005$	$(5.00\pm 0.14)\times 10^{6}$	$(4.26\pm 0.08)\times 10^{7}$	256.7
TINN	3663	$263\pm 7$	$74\pm 10$	$0.126\pm 0.004$	$(6.76\pm 0.08)\times 10^{6}$	$(5.30\pm 0.07)\times 10^{7}$	160.7
PINN	3631	$226\pm 4$	$67\pm 5$	$0.101\pm 0.002$	$(1.07\pm 0.02)\times 10^{7}$	$(7.27\pm 0.05)\times 10^{7}$	80.9
PI-DeepONet	3511	$643\pm 84$	$770\pm 135$	$0.107\pm 0.003$	$(8.39\pm 0.19)\times 10^{6}$	$(6.31\pm 0.09)\times 10^{7}$	99.8

Appendix

Table 5: Computational profiling of the final models. Training time and the optimization step attaining the minimum validation physics loss are recovered from the final training runs and reported as mean

\pm

std over six seeds. Inference is profiled post hoc from the restored best-validation checkpoints. For

B=1

, latency is the median forward-pass time; for larger batches, throughput is the batch size divided by the median forward-pass time. Peak allocation denotes the maximum JAX-managed device allocation during forward inference up to

B=10{,}000

Configuration	Params	Price MAE $\downarrow$	$\Delta$ MAE $\downarrow$	$K\Gamma$ MAE $\downarrow$	$u_{v}$ MAE $\downarrow$
$q=1$	1,630	$0.00769\pm 0.00110$	$0.01849\pm 0.00179$	$0.15218\pm 0.00999$	$0.00747\pm 0.00132$
$q=2$	2,285	$0.00777\pm 0.00084$	$0.01864\pm 0.00103$	$0.15013\pm 0.00562$	$0.00736\pm 0.00078$
$q=3$	2,940	$0.00799\pm 0.00088$	$0.01871\pm 0.00125$	$0.14326\pm 0.00687$	$0.00753\pm 0.00086$
$q=4$	3,595	$\mathbf{0.00625\pm 0.00095}$	$\mathbf{0.01605\pm 0.00201}$	$\mathbf{0.13615\pm 0.00919}$	$\mathbf{0.00582\pm 0.00116}$
$q=5$	4,250	$0.00736\pm 0.00090$	$0.01771\pm 0.00151$	$0.14396\pm 0.00673$	$0.00704\pm 0.00079$
$q=6$	4,905	$0.00696\pm 0.00124$	$0.01726\pm 0.00165$	$0.14460\pm 0.01120$	$0.00649\pm 0.00116$

Appendix

Table 6: Modulation-dimensionality ablation with fixed

(20,20)

structured backbone and

(12,10)

conditioner. Values are mean

\pm

std over six development seeds. Lower is better.

Backbone	Params	Price MAE $\downarrow$	$\Delta$ MAE $\downarrow$	$K\Gamma$ MAE $\downarrow$	$u_{v}$ MAE $\downarrow$
$(10,10)$ , fixed-budget diagnostic	1,385	$0.00845\pm 0.00098$	$0.01908\pm 0.00103$	$0.13680\pm 0.00795$	$0.00800\pm 0.00095$
$(12,12)$	1,915	$0.00758\pm 0.00110$	$0.01776\pm 0.00150$	$0.13925\pm 0.00589$	$0.00737\pm 0.00098$
$(16,16)$	2,675	$0.00687\pm 0.00175$	$0.01650\pm 0.00261$	$\mathbf{0.13192\pm 0.00451}$	$0.00636\pm 0.00181$
$(20,20)$	3,595	$\mathbf{0.00625\pm 0.00095}$	$\mathbf{0.01605\pm 0.00201}$	$0.13615\pm 0.00919$	$\mathbf{0.00582\pm 0.00116}$

Appendix

Table 7: Backbone-capacity ablation at fixed modulation dimensionality

q=4

. Values are mean

\pm

std over six development seeds. The

1{,}385

-parameter configuration is a fixed-budget diagnostic that also uses a smaller

(6,6)

conditioner; the clean fixed-conditioner capacity sweep spans

1{,}915

–

3{,}595

parameters. Lower is better.

Method	Params	Price MAE $\downarrow$	$\Delta$ MAE $\downarrow$	$K\Gamma$ MAE $\downarrow$	$u_{v}$ MAE $\downarrow$
TINN	2,727	$0.00794\pm 0.00105$	$0.0181\pm 0.0009$	$0.1513\pm 0.0039$	$0.00770\pm 0.00118$
PINN	2,623	$0.00847\pm 0.00100$	$0.0190\pm 0.0011$	$0.1475\pm 0.0045$	$0.00801\pm 0.00086$
JointResidual	2,665	$0.00797\pm 0.00168$	$0.0182\pm 0.0021$	$0.1482\pm 0.0103$	$0.00751\pm 0.00151$
BeTINN $(16{\times}16)$	2,675	$\mathbf{0.00687\pm 0.00175}$	$\mathbf{0.0165\pm 0.0026}$	$\mathbf{0.1319\pm 0.0045}$	$\mathbf{0.00636\pm 0.00181}$

Appendix

Table 8: Capacity-controlled development comparison around the

2{,}675

-parameter BeTINN configuration. Values are mean

\pm

std over six development seeds. Controls are independently scaled while preserving their native architectural geometry. Lower is better.

Method	Params	Price MAE $\downarrow$	$\Delta$ MAE $\downarrow$	$K\Gamma$ MAE $\downarrow$	$u_{v}$ MAE $\downarrow$
TINN	3,663	$0.00759\pm 0.00050$	$0.0179\pm 0.0009$	$0.148\pm 0.008$	$0.00725\pm 0.00067$
PINN	3,631	$0.00859\pm 0.00122$	$0.0187\pm 0.0014$	$0.149\pm 0.007$	$0.00816\pm 0.00116$
JointResidual	3,577	$0.00774\pm 0.00146$	$0.0182\pm 0.0020$	$0.151\pm 0.009$	$0.00741\pm 0.00148$
BeTINN	3,595	$\mathbf{0.00625\pm 0.00095}$	$\mathbf{0.0161\pm 0.0020}$	$\mathbf{0.136\pm 0.009}$	$\mathbf{0.00582\pm 0.00116}$

Appendix

Table 9: Capacity-controlled development comparison at the final BeTINN scale. Values are mean

\pm

std over six development seeds. This table reports development evidence and is distinct from the frozen final benchmark.

Configuration	Params	Mismatch (%)	Price MAE	$\Delta$ MAE	$K\Gamma$ MAE	$u_{v}$ MAE
BeTINN (reference)	3595	0.00	$\mathbf{0.00625\pm 0.00095}$	$\mathbf{0.01605\pm 0.00201}$	$\mathbf{0.1361\pm 0.0092}$	$\mathbf{0.00582\pm 0.00116}$
TINN	3663	1.89	$0.00759\pm 0.00050$	$0.01788\pm 0.00092$	$0.1479\pm 0.0077$	$0.00725\pm 0.00067$
TINN + parameter conditioning	3595	0.00	$0.00823\pm 0.00147$	$0.01857\pm 0.00229$	$0.1491\pm 0.0109$	$0.00774\pm 0.00142$
TINN + modified-MLP gating	3625	0.83	$0.00785\pm 0.00075$	$0.01809\pm 0.00053$	$0.1605\pm 0.0066$	$0.00764\pm 0.00062$
TINN + highway gating	3695	2.78	$0.00812\pm 0.00108$	$0.01842\pm 0.00128$	$0.1499\pm 0.0101$	$0.00783\pm 0.00104$
TINN + adaptive residual	3665	1.95	$0.00799\pm 0.00078$	$0.01869\pm 0.00106$	$0.1596\pm 0.0066$	$0.00795\pm 0.00074$

Appendix

Table 10: Capacity architectural component ablations on the six development seeds. Controlled variants are scaled approximately to the final BeTINN parameter budget while preserving their architectural structure. Values are mean

\pm

std. Lower is better.

Configuration	Params	Price diff.	$\Delta$ diff.	$K\Gamma$ diff.	$u_{v}$ diff.
TINN	3663	$+0.00133$ (1/6)	$+0.00182$ (2/6)	$+0.0118$ (2/6)	$+0.00143$ (1/6)
TINN + parameter conditioning	3595	$+0.00197$ (0/6)	$+0.00251$ (0/6)	$+0.0129$ (0/6)	$+0.00192$ (0/6)
TINN + modified-MLP gating	3625	$+0.00159$ (1/6)	$+0.00204$ (1/6)	$+0.0243$ (0/6)	$+0.00182$ (0/6)
TINN + highway gating	3695	$+0.00186$ (1/6)	$+0.00237$ (1/6)	$+0.0137$ (2/6)	$+0.00201$ (1/6)
TINN + adaptive residual	3665	$+0.00173$ (1/6)	$+0.00264$ (1/6)	$+0.0234$ (0/6)	$+0.00213$ (1/6)
JointResidual	3577	$+0.00149$ (1/6)	$+0.00217$ (1/6)	$+0.0151$ (1/6)	$+0.00160$ (1/6)

Appendix

Table 11: Paired effects of final-capacity architectural ablations relative to BeTINN over the six common development seeds. Differences are computed seed-wise as

\mathrm{ablation}-\textsc{BeTINN}{}

, so positive values indicate larger error for the ablation. Parentheses report the number of seeds on which the ablation attains lower error than BeTINN .

Baseline	Metric	Mean diff.	Median diff.	BeTINN wins
TINN	Price	$-0.00164$	$-0.00118$	6/6
	$\Delta$	$-0.00177$	$-0.00159$	5/6
	$K\Gamma$	$-0.0149$	$-0.0157$	6/6
	$u_{v}$	$-0.00141$	$-0.00111$	6/6
PINN	Price	$-0.00110$	$-0.000959$	5/6
	$\Delta$	$-0.000783$	$-0.000504$	4/6

Appendix

Table 12: Paired final-benchmark differences between BeTINN and each main baseline over the six common confirmatory seeds. Differences are computed seed-wise as

\textsc{BeTINN}{}-\mathrm{baseline}

, so negative values indicate lower BeTINN error. “Wins” denotes the number of seeds on which BeTINN attains lower error.

Metric	Method	Price	$\Delta$	$K\Gamma$	$u_{v}$	$2\sqrt{v}\,u_{v}$	$-u_{\tau}$	$u_{r}$
MAE	BeTINN	$0.00746\pm 0.00097$	$0.0180\pm 0.0010$	$0.143\pm 0.005$	$0.00712\pm 0.00089$	$0.00942\pm 0.00114$	$0.00475\pm 0.00063$	$0.0597\pm 0.0151$
	TINN	$0.00910\pm 0.00062$	$0.0197\pm 0.0009$	$0.158\pm 0.006$	$0.00853\pm 0.00046$	$0.0113\pm 0.0006$	$0.00566\pm 0.00027$	$0.0808\pm 0.0118$
	PINN	$0.00857\pm 0.00054$	$0.0188\pm 0.0005$	$0.149\pm 0.003$	$0.00800\pm 0.00049$	$0.0106\pm 0.0007$	$0.00528\pm 0.00030$	$0.0817\pm 0.0090$
	PI-DeepONet	$0.00974\pm 0.00063$	$0.0232\pm 0.0007$	$0.199\pm 0.016$	$0.0109\pm 0.0005$	$0.0145\pm 0.0007$	$0.00709\pm 0.00028$	$0.0774\pm 0.0076$
RMSE	BeTINN	$0.0124\pm 0.0011$	$0.0382\pm 0.0009$	$0.397\pm 0.022$	$0.0123\pm 0.0014$	$0.0148\pm 0.0017$	$0.00906\pm 0.00150$	$0.106\pm 0.027$
	TINN	$0.0148\pm 0.0009$	$0.0412\pm 0.0014$	$0.423\pm 0.021$	$0.0144\pm 0.0006$	$0.0174\pm 0.0008$	$0.0101\pm 0.0008$	$0.155\pm 0.025$

Appendix

Table 13: Global pricing and sensitivity errors on the frozen final evaluation panel. Values are mean

\pm

std over six seeds. Lower is better.

Error metric	Spearman
Price MAE	0.578
$\Delta$ MAE	0.682
$K\Gamma$ MAE	0.617
$u_{v}$ MAE	0.698

Appendix

Table 14: Spearman association between the best validation physics loss and downstream prediction error across the

24

final model–seed pairs (

4

architectures

\times

6

confirmatory seeds). Positive values indicate that runs with larger validation physics loss tend to exhibit larger downstream error.

	Price MAE				$\Delta$ MAE
Region	BeTINN	TINN	PINN	PI-DeepONet	BeTINN	TINN	PINN	PI-DeepONet
ATM (501)	.00672	.00934	.00839	.0106	.00992	.0110	.0109	.0141
ATM tight (253)	.00702	.00987	.00890	.0112	.00978	.0110	.0109	.0135
ATM wide (1297)	.00700	.00959	.00860	.0111	.00992	.0110	.0109	.0139
ATM wide & short maturity (62)	.00432	.00417	.00406	.00623	.0174	.0184	.0194	.0306
ATM & near maturity (8)	.00559	.00479	.00507	.00520	.0318	.0332	.0387	.0741

Appendix

Table 15: Region-wise mean MAE for Price and

\Delta

on the frozen final evaluation panel. The number of evaluation points in each predefined subset is shown in parentheses. Bold indicates the lowest mean error within each region and metric. All regions are fixed before model-level aggregation.

	$K\Gamma$ MAE				$u_{v}$ MAE
Region	BeTINN	TINN	PINN	PI-DeepONet	BeTINN	TINN	PINN	PI-DeepONet
ATM (501)	.0345	.0398	.0390	.0561	.0102	.0124	.0116	.0174
ATM tight (253)	.0328	.0398	.0384	.0561	.0101	.0123	.0115	.0171
ATM wide (1297)	.0339	.0392	.0383	.0590	.0100	.0124	.0115	.0172
ATM wide & short maturity (62)	.188	.181	.182	.307	.0131	.0138	.0133	.0246
ATM & near maturity (8)	.536	.402	.376	.476	.00849	.0110	.0114	.0148

Appendix

Table 16: Region-wise mean MAE for strike-scaled Gamma

K\Gamma

and variance sensitivity

u_{v}

Family	Candidate	Depth	Widths	Params	Mismatch (%)	Price	$\Delta$	$K\Gamma$	$u_{v}$	Ranks / mean rank
TINN	Capacity-matched	2	$(37,37)$	3663	1.89	0.007587	0.0179	0.148	0.00725	2/2/1/1 1.50
TINN	Funnel	2	$(50,25)$	3585	0.28	0.007585	0.0175	0.150	0.00734	1/1/2/2 1.50
TINN	Depth 3	3	$(26,26,26)$	3483	3.12	0.009363	0.0201	0.158	0.00873	3/3/3/3 3.00
PINN	Capacity-matched	2	$(55,55)$	3631	1.00	0.00859	0.01868	0.1489	0.00816	1/2/3/1 1.75
PINN	Depth 3	3	$(39,39,39)$	3511	2.34	0.00997	0.02016	0.1448	0.00900	3/3/1/3 2.50
PINN	Depth 4	4	$(33,33,33,33)$	3697	2.84	0.00893	0.01866	0.1451	0.00821	2/1/2/2 1.75

Appendix

Table 17: Architecture sensitivity of the TINN and PINN baselines under approximately matched parameter budgets. All candidates are evaluated on the same six development seeds.

Method	Params	Price MAE	$\Delta$ MAE	$K\Gamma$ MAE	$u_{v}$ MAE
DeepSVM-adapted	33920	0.1258 $\pm$ 0.0004	0.2073 $\pm$ 0.0003	0.376 $\pm$ 0.002	0.0567 $\pm$ 0.0004
BeTINN	3595	0.00746 $\pm$ 0.00097	0.0180 $\pm$ 0.0010	0.143 $\pm$ 0.005	0.00712 $\pm$ 0.00089
TINN	3663	0.00910 $\pm$ 0.00062	0.0197 $\pm$ 0.0009	0.158 $\pm$ 0.006	0.00853 $\pm$ 0.00046
PINN	3631	0.00857 $\pm$ 0.00054	0.0188 $\pm$ 0.0005	0.149 $\pm$ 0.003	0.00800 $\pm$ 0.00049
PI-DeepONet	3511	0.00974 $\pm$ 0.00063	0.0232 $\pm$ 0.0007	0.199 $\pm$ 0.016	0.0109 $\pm$ 0.0005

Appendix

Table 18: External adapted DeepSVM reference. All models are evaluated on the same frozen

10{,}000

-point panel against the same semi-analytic oracle. Values are mean

\pm

std over seeds.

Figure 5: Local Delta and Gamma profiles and pointwise errors. Delta

\Delta

, absolute Delta error, strike-scaled Gamma

K\Gamma

, and absolute Gamma error are shown as functions of log-moneyness across representative interior, near-maturity, and stress regimes. Error panels use logarithmic scales to expose localized discrepancies spanning several orders of magnitude.

Figure 6: Variance-sensitivity profiles across representative regimes. The variance sensitivity and its absolute error are shown as functions of log-moneyness for the representative interior, near-maturity, and stress regimes. The error panels use a logarithmic scale to resolve differences across several orders of magnitude.

Figure 7: Spatial distribution of Delta error. Absolute pointwise Delta error, $|\hat{\Delta}-\Delta^{\star}|$ , is shown over the $(x,v)$ plane across representative interior, near-maturity, and stress regimes. All methods are evaluated on the same $55\times 55$ grid, and all panels share a common logarithmic color scale. Narrow low-error structures correspond to locations of small pointwise error.

Figure 8: Spatial distribution of strike-scaled Gamma error. Absolute pointwise error in $K\Gamma$ is shown over the $(x,v)$ plane across representative interior, near-maturity, and stress regimes. All panels use the same $55\times 55$ evaluation grid and a shared logarithmic color scale. Narrow low-error structures correspond to locations where the pointwise error approaches zero.

Figure 9: Spatial distribution of variance-sensitivity error. Absolute pointwise variance-sensitivity error, $|\hat{u}_{v}-u_{v}^{\star}|$ , is shown over the $(x,v)$ plane across representative interior, near-maturity, and stress regimes. All panels use the same $55\times 55$ evaluation grid and a shared logarithmic color scale.

Figure 10: Spatial distribution of the Heston PDE residual. Absolute pointwise PDE residuals are shown over the $(x,v)$ plane across representative interior, near-maturity, and stress regimes. Residuals are evaluated on the same $55\times 55$ grids used for the spatial derivative diagnostics and displayed using a common logarithmic color scale.

Figure 11: Extended sensitivity profiles across representative regimes. Normalized calendar Theta

-u_{\tau}

and interest-rate sensitivity

u_{r}

are shown as functions of log-moneyness

x=\log(S/K)

for representative interior, near-maturity, and stress regimes.

Figure 12: Shape-constraint violation rates. Percentage of points on the frozen confirmatory panel violating the primary structural conditions using tolerance

\varepsilon=10^{-4}

. The corresponding semi-analytic oracle has zero violations under the same tolerance. Lower values indicate fewer violations.

PDE	Model	Params	Rel. $L_{2}$	MAE
Parametric Burgers	PINN	3017	$0.04811\pm 0.0560$	$0.003818\pm 0.0062$
	TINN	2985	$0.003053\pm 0.0035$	$0.000538\pm 0.0011$
	BeTINN	2967	$0.000379\pm 0.00023$	$0.00000825\pm 0.0000037$
Parametric Wave	PINN	3017	$0.002724\pm 0.0035$	$0.0000453\pm 0.000049$
	TINN	2985	$0.002101\pm 0.0021$	$0.0000717\pm 0.000076$
	BeTINN	2967	$0.004134\pm 0.0056$	$0.0000733\pm 0.000089$

Appendix

Table 19: Errors over the additional frozen parametric PDE evaluation panels, reported as mean

\pm

std over six seeds.

Neural PDE Solvers

Neural PDE solvers use trainable models to approximate solutions or solution operators for partial differential equations, often with physical constraints.

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Jun 25, 2026

Error-Conditioned Neural Solvers

Neural surrogate models offer fast approximate mappings from PDE parameters to solutions, but they typically treat solving as a purely statistical task: once trained, they struggle to correct their own constraint violations and extrapolate beyond the training distribution. Recent hybrid methods promote physical correctness by targeting the PDE residual via gradient descent or Gauss--Newton steps, but inherit the compute cost and instability of the underlying classical optimizers. We show, theoretically and empirically, that numerically minimizing the PDE residual can be an unreliable proxy for reconstruction accuracy in ill-conditioned systems, explaining why these methods often do not make accurate predictions despite achieving low residuals. We propose error-conditioned Neural Solvers (ENS), built on a different principle: rather than an optimization target, the PDE residual field is passed as a direct input to the network at each iteration, enabling it to read the spatial structure of its own errors and learn an update policy to iteratively correct its predictions. Across four PDE families, ENS attains the highest prediction accuracy in the large majority of settings, with gains reaching

10\times

on turbulent Kolmogorov flow, while avoiding the expensive compute cost of hybrid methods. ENS's learned correction policy generalizes under distribution shift, including zero-shot parameter changes and cross-equation transfer, where its relative advantage is largest in the ill-conditioned regimes where residual minimization is least reliable. Project website: https://neuralsolver.github.io/.

Haina Jiang, Liam Wang, Peng-Chen Chen +4

University of Michigan · KAIST AI · Los Alamos National Laboratory

Neural PDE Solvers PDE Surrogate Modeling

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May 29, 2026

Oscillatory State-Space Models as Inductive Biases for Physics-Informed Neural PDE Solvers

Solving time-dependent partial differential equations (PDEs) is an important problem in computational science and engineering. Physics-informed neural networks (PINNs) learn PDE solutions from governing equations. However, accurately capturing temporal evolution remains challenging. Recent sequence-model-based approaches parameterize time evolution using general-purpose sequence models, which capture temporal dependencies but do not explicitly encode the structured dynamics of PDE solutions. In addition, their memory requirements can scale unfavorably with sequence length and resolution, limiting applicability in large-scale or high-dimensional settings. This work introduces a PINN approach that incorporates oscillatory state-space dynamics to represent the modal structure of PDE solutions. The proposed method leverages a linear-oscillator-based temporal evolution, together with a PDE-aware spectral basis in space. This design enables closed-form spatial differentiation and consistent enforcement of boundary conditions. The method is evaluated on forward, inverse, and high-dimensional PDE problems, including cases up to 100 spatial dimensions. The results show improved accuracy and reduced memory usage compared to recent sequence-model-based PINN approaches. Overall, this work highlights the benefits of incorporating structured dynamical priors into the temporal evolution of neural PDE solvers and suggests designing more physics-aligned and computationally efficient PINN architectures.

Abhishek Chandra, Taniya Kapoor

KTH Royal Institute of Technology Stockholm, Sweden · Wageningen University & Research Wageningen, The Netherlands

Neural PDE Solvers Physics-Informed ML PDE Inverse Problems State Space Models

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Jun 20, 2026

Parameterized Representations via Implicit Stochastic Modulation for High-Dimensional and High-Order Neural PDE Solvers

Solving high-dimensional and high-order PDEs is challenged by the coupled growth of spatial dimensionality and derivative order. Recent stochastic derivative estimators reduce this cost by replacing full derivative tensors with randomized dimension or Taylor estimators, but they are mostly designed for fixed physical parameters and require retraining for each new parameter. We show that direct conditional parameterization of such solvers entangles physical parameters with the high-order automatic differentiation graph, causing extra memory growth and parameter-induced variance amplification. We propose Parameterized Representations via Implicit Stochastic Modulation (PRISM), a plug-and-play framework for parameterized high-dimensional and high-order stochastic neural PDE solvers. PRISM uses a hyper-generator to map physical parameters to affine modulators that scale and shift a purely spatial latent manifold, while keeping parameter branches value-connected but spatial-tangent-disconnected. This design preserves unbiased stochastic dimension and Taylor estimators, removes the parameter encoder from high-order spatial AD, and provides a variance-aware Lipschitz envelope over the parameter space. We prove parameterized unbiasedness, estimation-error bounds, and convergence under bounded stochastic variance. Experiments with PRISM-STDE and PRISM-SDGD on nonlinear parameterized PDEs show stable zero-shot generalization, reduced memory usage, and scalability up to 100,000 dimensions on a single GPU, with efficient low-rank SVD adaptation for unseen parameters.

Zhangyong Liang, Huanhuan Gao

National Center for Applied Mathematics, Tianjin University, Tianjin, 300072, PR China · School of Mechanical and Aerospace Engineering, Jilin University, Changchun, 130025, PR China

Neural PDE Solvers PDE Solving

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Abstract

Neural PDE Solvers

Explore similar work

Error-Conditioned Neural Solvers

Oscillatory State-Space Models as Inductive Biases for Physics-Informed Neural PDE Solvers

Parameterized Representations via Implicit Stochastic Modulation for High-Dimensional and High-Order Neural PDE Solvers