When Does Cheap Physics Help? Gated Multi-Fidelity Physics-Informed Networks with Provable Sample Savings and Conformal Guarantees

laf Yunus Laitinen Imanov; Rana Irem Turhan; Isgandar Jafarli

cs.LGFeb 1, 2026

When Does Cheap Physics Help? Gated Multi-Fidelity Physics-Informed Networks with Provable Sample Savings and Conformal Guarantees

Authors: laf Yunus Laitinen Imanov, Rana Irem Turhan, Isgandar Jafarli

Organizations: Department of Applied Mathematics and Computer Science (DTU Compute), Technical University of Denmark, Kongens Lyngby, Denmark

Abstract

High-fidelity (HF) simulations of parametric partial differential equations are expensive, whereas low-fidelity (LF) models such as coarse grids or simplified physics are cheap but biased. Multi-fidelity physics-informed neural networks combine the two, yet it remains unclear when cheap physics actually reduces the number of HF solves, and existing methods silently fail when the LF model is misleading and offer no guaranteed uncertainty. We answer the first question with a sharp sample-complexity bound: learning the HF field through its discrepancy from the LF model reduces the number of HF samples needed for accuracy $\varepsilon$ by the factor $κ^{2γ}$ , where $κ$ is the ratio of the complexity of the discrepancy to that of the HF field and $γ$ is the effective-dimension exponent, and we confirm the predicted scaling with exact risk computations in a kernel model. Guided by the analysis, we propose GaMF-PINN, a gated multi-fidelity PINN that learns where to trust the LF model and falls back to physics-only learning where it does not, together with a function-level conformal procedure that returns uncertainty bands containing the entire HF solution with probability at least $1-α$ in finite samples. We release MFPDE-Bench, an open benchmark of paired LF/HF solvers for four parametric PDEs with coarse-grid and simplified-physics LF models. Across MFPDE-Bench, GaMF-PINN reaches the accuracy of a standard HF-PINN with up to 8x fewer HF solves, remains robust when the LF model is deliberately corrupted, and attains the nominal coverage where Bayesian PINNs and deep ensembles under-cover.

Figures & tables

Figure 1: Exact verification of Theorems 4.1 and 4.2 (Matérn-3/2 kernel,

\sigma=0.05

, risk computed in closed form from Equation 2 with oracle

\lambda

). (a) Exact risk (solid) and the upper bound (dotted) for direct learning and for residual learning with discrepancies of decreasing relative norm

\kappa

; the bound holds for every

N

and

\kappa

. (b) Ratio of HF samples needed to reach risk

\varepsilon=2\times 10^{-4}

. The bound-based ratio follows

\kappa^{2\gamma}

with the measured

\gamma=0.26

(the Sobolev prediction is

1/4

); the exact ratio shows the same monotone trend, with a smaller saving because the bound is a worst case over targets of a given norm.

Setting	HF time (s)	LF time (s)	Ratio	LF error (%)	$\hat{\kappa}$
Burgers	$11.24\pm 0.35$	$0.0078\pm 0.0003$	$1441\times$	$14.8\pm 1.4$	$0.07\pm 0.01$
Diffusion (coarse)	$4.85\pm 0.08$	$0.0015\pm 0.0001$	$3233\times$	$6.2\pm 0.7$	$0.018\pm 0.009$
Diffusion (simplified)	$4.85\pm 0.08$	$0.0240\pm 0.0006$	$202\times$	$38.8\pm 2.4$	$0.52\pm 0.03$
Allen–Cahn	$16.40\pm 0.38$	$0.0182\pm 0.0005$	$901\times$	$36.2\pm 3.5$	$0.60\pm 0.03$
Navier–Stokes	$52.60\pm 0.85$	$0.0345\pm 0.0009$	$1525\times$	$25.4\pm 2.1$	$0.12\pm 0.01$

Table 1: MFPDE-Bench statistics measured on the test split: mean wall-clock time per solve, cost ratio, mean relative

L^{2}

error of the LF solver, and an empirical proxy

\hat{\kappa}

\kappa

(discrete

H^{1}

-seminorm ratio of the discrepancy to the HF field).

Method	Burgers	Diff. (coarse)	Diff. (simpl.)	Allen–Cahn	Navier–Stokes
LF surrogate	$14.82\pm 0.18$	$6.24\pm 0.09$	$38.92\pm 0.45$	$36.40\pm 0.42$	$25.60\pm 0.31$
HF-PINN	$2.84\pm 0.22$	$3.15\pm 0.28$	$3.15\pm 0.28$	$5.42\pm 0.46$	$4.68\pm 0.35$
KO co-kriging	$3.45\pm 0.31$	$1.85\pm 0.16$	$6.12\pm 0.52$	$8.90\pm 0.72$	$6.30\pm 0.48$
MF-PINN (lin.+nonlin.)	$1.42\pm 0.12$	$0.98\pm 0.08$	$2.85\pm 0.24$	$3.85\pm 0.31$	$2.55\pm 0.20$
Residual MF-PINN	$1.28\pm 0.10$	$0.85\pm 0.07$	$2.95\pm 0.25$	$3.62\pm 0.29$	$2.38\pm 0.18$
GaMF-PINN (ours)	$\mathbf{0.86\pm 0.06}$	$\mathbf{0.58\pm 0.04}$	$\mathbf{1.45\pm 0.11}$	$\mathbf{2.15\pm 0.18}$	$\mathbf{1.48\pm 0.11}$

Table 2: Test relative

L^{2}

error (%) with

N_{\mathrm{HF}}=8

HF solves (mean

\pm

std over 5 seeds). Best in bold.

Figure 2: Test relative

L^{2}

error versus the number of HF solves (log–log; mean

\pm

std over 5 seeds). The dashed line is the LF surrogate.

Problem	Method	$A=0$	$A=0.5$	$A=1$	$A=2$
Burgers	HF-PINN (no LF)	$2.84\pm 0.22$	$2.84\pm 0.22$	$2.84\pm 0.22$	$2.84\pm 0.22$
	Residual MF-PINN	$1.28\pm 0.10$	$3.42\pm 0.28$	$6.85\pm 0.54$	$14.20\pm 1.12$
	MF-PINN (lin.+nonlin.)	$1.42\pm 0.12$	$3.18\pm 0.25$	$5.92\pm 0.48$	$11.80\pm 0.95$
	GaMF-PINN (ours)	$\mathbf{0.86\pm 0.06}$	$\mathbf{1.75\pm 0.14}$	$\mathbf{2.70\pm 0.21}$	$\mathbf{2.89\pm 0.22}$
Diffusion	HF-PINN (no LF)	$3.15\pm 0.28$	$3.15\pm 0.28$	$3.15\pm 0.28$	$3.15\pm 0.28$
	Residual MF-PINN	$0.85\pm 0.07$	$2.92\pm 0.24$	$6.40\pm 0.52$	$14.10\pm 1.15$

Table 3: Misleading-LF stress test (

N_{\mathrm{HF}}=8

): test relative

L^{2}

error (%) as the LF data is corrupted with amplitude

A

. The HF-PINN does not use LF data and is repeated for reference.

Problem	Method	Field cov. (%)	Pointwise cov. (%)	Width
All problems	Deep ensemble (Gaussian)	$37.5\pm 4.2$	$84.2\pm 2.0$	$0.44\pm 0.03$
	B-PINN (HMC)	$28.0\pm 3.5$	$80.8\pm 2.4$	$0.39\pm 0.04$
	KO co-kriging (GP)	$42.5\pm 4.6$	$86.5\pm 1.8$	$0.52\pm 0.04$
	Ensemble + conformal (ours)	$\mathbf{92.5\pm 2.2}$	$\mathbf{99.2\pm 0.3}$	$\mathbf{0.85\pm 0.05}$

Table 4: Uncertainty quantification at nominal level

90\%

(

N_{\mathrm{HF}}=8

n=20

calibration solves). Width is relative to the standard deviation of the test fields.

Variant	Burgers	Diff. (coarse)	Diff. (simpl.)	Allen–Cahn	Navier–Stokes
GaMF-PINN (full)	$\mathbf{0.86\pm 0.06}$	$\mathbf{0.58\pm 0.04}$	$\mathbf{1.45\pm 0.11}$	$\mathbf{2.15\pm 0.18}$	$\mathbf{1.48\pm 0.11}$
w/o gate ( $a\equiv 1$ )	$1.15\pm 0.09$	$0.74\pm 0.06$	$2.65\pm 0.22$	$3.24\pm 0.26$	$2.12\pm 0.16$
w/o gate and $\rho$ (residual)	$1.28\pm 0.10$	$0.85\pm 0.07$	$2.95\pm 0.25$	$3.62\pm 0.29$	$2.38\pm 0.18$
w/o physics residual	$2.95\pm 0.24$	$2.10\pm 0.18$	$4.85\pm 0.40$	$7.40\pm 0.58$	$5.12\pm 0.40$
w/o LF (HF-PINN)	$2.84\pm 0.22$	$3.15\pm 0.28$	$3.15\pm 0.28$	$5.42\pm 0.46$	$4.68\pm 0.35$
w/o LF, w/o physics	$8.45\pm 0.68$	$9.80\pm 0.75$	$9.80\pm 0.75$	$18.25\pm 1.40$	$14.60\pm 1.15$

Table 5: Ablations at

N_{\mathrm{HF}}=8

: test relative

L^{2}

error (%).

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Item	Value
Corrector $\delta_{\theta}$	4 layers $\times$ 64, tanh
Gate $g_{\theta}$ , scale $r_{\theta}$	2 layers $\times$ 32, tanh
LF surrogate $\hat{u}_{\mathrm{LF}}$	5 layers $\times$ 128, tanh
Fourier harmonics (periodic coordinates)	4
Optimizer [ Kingma and Ba, 2015 ]	Adam, lr $10^{-3}$ , cosine
Iterations	$2\times 10^{4}$

Appendix

Table 6: Hyperparameters shared by all neural models and baselines.

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