Manifold-Informed Neural Networks for Low-Label ECG Electrolyte Inference

Juntang Wang; Hao Wu; Yihan Wang; Dongmian Zou; Shixin Xu

cs.LGOct 20, 2026

Manifold-Informed Neural Networks for Low-Label ECG Electrolyte Inference

Authors: Juntang Wang, Hao Wu, Yihan Wang, Dongmian Zou, Shixin Xu

Organizations: Duke University, USA

Abstract

Physics-informed neural networks (PINNs) supervise on the residual of a known differential operator. Many high-stakes inference problems admit no such operator: clinical signals, behavioral measurements, materials properties. We introduce the Manifold-Informed Neural Network (MINN), a PINN-parallel paradigm that replaces the physics operator residual with a manifold operator residual: a discrete Poisson equation $\mathbf{L}_G \mathbf{H} = s_φ(\mathbf{x}, \mathbf{m})$ on a $k$ -nearest-neighbor graph in the learned latent space, with a learned, task-aware source. The construction is a structural parallel to PINNs (squared operator residual plus label data, in place of operator residual plus boundary data); the $s{=}0$ limit recovers the homogeneous Laplace equation $\mathbf{L}_G\mathbf{H}=\mathbf{0}$ (classical manifold regularization solves the same equation but minimizes the Dirichlet energy $\operatorname{tr}(\mathbf{H}^\top\mathbf{L}_G\mathbf{H})$ , a different penalty form). The source $s_φ$ takes input features and auxiliary metadata directly, breaking the homogeneous smoothness prior that limits classical Laplacian penalties; gated Feature-wise Linear Modulation separately fuses metadata into the prediction branch. We evaluate MINN on three-class classification of potassium, calcium, and magnesium from 12-lead ECGs in MIMIC-IV-ECG, against twelve baselines spanning supervised and tabular methods, ECG foundation models, and manifold/graph approaches. MINN attains the highest K $^+$ Macro-F1 of 0.61 (best baseline 0.55), with consistent gains on Ca $^{2+}$ and Mg $^{2+}$ and a 31% relative improvement at 1% labeled data. Six targeted ablations show that the inhomogeneous Poisson formulation, not graph smoothing alone, drives the gain.

Figures & tables

Panel of Figure 1 — Figure 1: Overview of the MINN architecture. The input signal is mapped to a latent embedding by the encoder $h_{\theta}$ , fused with auxiliary metadata via gated FiLM, and passed to task-specific prediction heads ( Prediction Branch ). Simultaneously, a $k$ -NN graph in the latent space and a learned source network $s_{\phi}$ define the manifold operator residual ( Manifold Branch ). Dashed arrows indicate gradient flow from the two loss terms.

Supervised & Tabular
	K +		Ca 2+		Mg 2+
Method	F1 ( $\uparrow$ )	BAcc ( $\uparrow$ )	F1 ( $\uparrow$ )	BAcc ( $\uparrow$ )	F1 ( $\uparrow$ )	BAcc ( $\uparrow$ )
ResNet-18 [ 36 ]	0.48 $\pm.03$	0.55 $\pm.03$	0.40 $\pm.04$	0.46 $\pm.04$	0.32 $\pm.05$	0.39 $\pm.05$
Logit Adj. [ 32 ]	0.53 $\pm.02$	0.60 $\pm.02$	0.43 $\pm.02$	0.50 $\pm.02$	0.34 $\pm.03$	0.41 $\pm.02$
TabPFN [ 34 ]	0.47 $\pm.02$	0.54 $\pm.02$	0.43 $\pm.01$	0.50 $\pm.02$	0.36 $\pm.02$	0.43 $\pm.02$
CatBoost [ 35 ]	0.48 $\pm.02$	0.55 $\pm.02$	0.44 $\pm.01$	0.51 $\pm.02$	0.37 $\pm.02$	0.47 $\pm.02$

Table 1: Main results on MIMIC-IV-ECG electrolyte classification (100% labeled data). Macro-F1 and Balanced Accuracy (BAcc) are reported as mean

\pm

std over 5 seeds. Best results in bold .

Panel of Figure 2 — Figure 2: (a) K + Macro-F1 as a function of labeled data fraction. MINN maintains the largest advantage at low fractions (1%–10%), where the manifold operator residual compensates for label scarcity. The shaded band shows $\pm$ 1 std for MINN. (b) Mean source norm $\lVert{\bm{S}}_{i}\rVert_{2}$ by K + class ( $\pm$ 1 std); minority classes (hypo, hyper) exhibit larger source norms than the majority class, suggesting the source compensates for label imbalance.

Variant	K + ( $\uparrow$ )	Ca 2+ ( $\uparrow$ )	Mg 2+ ( $\uparrow$ )
MINN (full)	0.61 $\pm.02$	0.50 $\pm.02$	0.40 $\pm.03$
$-$ Operator residual ( $\lambda{=}0$ )	0.54 $\pm.03$	0.44 $\pm.03$	0.35 $\pm.03$
$-$ Source ( $s_{\phi}{=}0$ )	0.56 $\pm.02$	0.46 $\pm.03$	0.36 $\pm.04$
$-$ Poisson $\to$ Classical reg.	0.55 $\pm.03$	0.45 $\pm.03$	0.36 $\pm.03$
$-$ Learned source $\to$ Label-only	0.57 $\pm.02$	0.45 $\pm.03$	0.36 $\pm.03$
$-$ Latent $\to$ Logit constraint	0.56 $\pm.03$	0.46 $\pm.03$	0.37 $\pm.03$

Table 2: Ablation study results (Macro-F1

\pm

std, 100% labeled data, 5 seeds). Best in bold .

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Description
${\bm{x}}\in\mathbb{R}^{D}$	Input signal
${\bm{m}}\in\mathbb{R}^{p}$	Auxiliary metadata vector
$y\in\{1,\dots,B\}$	Categorical label ( $B$ classes)
$N,\;T,\;d$	Training samples, prediction tasks, embedding dimension
$h_{\theta}:\mathcal{X}\to\mathbb{R}^{d}$	Encoder network (parameters $\theta$ )
${\bm{h}},\;\tilde{{\bm{h}}}$	Latent embedding $h_{\theta}({\bm{x}})$ / metadata-fused embedding

Appendix

Table 3: Summary of notation used in Section 3 .

Property	Manifold Reg. [ 9 ]	MINN
Penalty form	$\operatorname{tr}({\bm{H}}^{\top}{\bm{L}}_{G}{\bm{H}})$ (Dirichlet)	$\lVert{\bm{L}}_{G}{\bm{H}}-{\bm{S}}\rVert_{2}^{2}$ (squared residual)
PDE analogy	Laplace ( $\Delta u=0$ )	Poisson ( $\Delta u=f$ )
Source term	None (zero RHS)	Learned $s_{\phi}({\bm{x}},{\bm{m}})$
Task dependence	No	Yes (source encodes metadata)
Collocation analog	No	Yes (residual at all nodes)
Low-label stability	Degenerates [ 10 ]	Stable (non-trivial source)

Appendix

Table 4: Structural comparison of classical manifold regularization and MINN. Note that the underlying PDE and the optimized penalty are distinct: classical manifold regularization solves the homogeneous Laplace equation

{\bm{L}}_{G}{\bm{H}}={\bm{0}}

via the Dirichlet penalty

\operatorname{tr}({\bm{H}}^{\top}{\bm{L}}_{G}{\bm{H}})

; MINN solves the inhomogeneous Poisson equation

{\bm{L}}_{G}{\bm{H}}={\bm{S}}

via the squared-residual penalty

\lVert{\bm{L}}_{G}{\bm{H}}-{\bm{S}}\rVert_{2}^{2}

. Setting

{\bm{S}}={\bm{0}}

in MINN recovers the homogeneous Laplace equation but with a squared-Laplacian (biharmonic) penalty, which is a different functional from the Dirichlet penalty.

Panel of Figure 3 — Figure 3: Data processing pipeline. Raw 12-lead ECGs undergo R-peak detection and multi-beat segmentation, then are paired with the closest laboratory blood draw within a $\pm$ 1-hour window. Clinical metadata (demographics and active medications) are extracted and joined.

Panel of Figure 4 — Figure 4: Structural parallel between PINNs and MINN. MINN replaces the known PDE operator with the data-driven graph Laplacian and the known source with a learned, task-dependent source network, while preserving the operator residual formulation.

Beats per window ( $N$ )	K + Macro-F1 ( $\uparrow$ )
$N=1$ (single beat)	0.54 $\pm.04$
$N=3$	0.59 $\pm.03$
$N=5$ ( default )	0.61 $\pm.02$
$N=8$	0.60 $\pm.03$
$N=12$	0.58 $\pm.04$

Appendix

Table 5: Beat-window sensitivity analysis: K + Macro-F1 (mean

\pm

std, 5 seeds) across input segmentation lengths

N

. Default highlighted in bold .

$k$ sweep ( $\lambda{=}0.1$ )				$\lambda$ sweep ( $k{=}15$ )
$k=5$	$k=10$	$k=15$	$k=30$	$\lambda{=}0.01$	$\lambda{=}0.1$	$\lambda{=}0.3$	$\lambda{=}1.0$
0.57 $\pm.04$	0.60 $\pm.02$	0.61 $\pm.02$	0.59 $\pm.03$	0.58 $\pm.03$	0.61 $\pm.02$	0.59 $\pm.03$	0.55 $\pm.05$

Appendix

Table 6: K + Macro-F1 sensitivity to neighbor count

k

(at

\lambda{=}0.1

) and constraint weight

\lambda

(at

k{=}15

), 100% labeled data, 5 seeds. Defaults in bold .

Encoder	w/ MINN	w/o MINN	$\Delta$ ( $\uparrow$ )
ResNet-18	0.61 $\pm.02$	0.54 $\pm.02$	+0.07
4-layer CNN	0.55 $\pm.02$	0.48 $\pm.02$	+0.07

Appendix

Table 7: K + Macro-F1 with and without the manifold operator residual for two encoder architectures (mean

\pm

std, 5 seeds).

Electrolyte	Bin	Train	Val	Test
K +	Hypo	2,187 (1,102)	274 (138)	273 (137)
	Normal	8,280 (4,172)	1,035 (522)	1,033 (521)
	Hyper	1,123 (566)	141 (71)	139 (70)
Ca 2+	Hypo	1,616 (814)	202 (102)	201 (101)
	Normal	7,465 (3,762)	933 (470)	932 (470)
	Hyper	965 (486)	121 (61)	119 (60)

Appendix

Table 8: Dataset statistics: recordings (unique patients) per electrolyte bin after lab matching. Not all recordings have lab results for all three electrolytes.

		K +			Ca 2+			Mg 2+
Stratum	$n$	MINN	RN+FiLM	best base	MINN	RN+FiLM	best base	MINN	RN+FiLM	best base
Male	${\sim}9$ k	0.61 $\pm.02$	0.56 $\pm.03$	0.55 $\pm.02$	0.50 $\pm.02$	0.46 $\pm.03$	0.45 $\pm.03$	0.40 $\pm.03$	0.37 $\pm.04$	0.36 $\pm.03$
Female	${\sim}9$ k	0.61 $\pm.03$	0.57 $\pm.03$	0.55 $\pm.02$	0.49 $\pm.03$	0.46 $\pm.03$	0.44 $\pm.03$	0.39 $\pm.03$	0.36 $\pm.04$	0.35 $\pm.03$
Age $<50$	${\sim}5$ k	0.62 $\pm.02$	0.58 $\pm.03$	0.55 $\pm.02$	0.51 $\pm.03$	0.47 $\pm.03$	0.45 $\pm.03$	0.41 $\pm.03$	0.38 $\pm.04$	0.36 $\pm.03$
Age $50$ – $70$	${\sim}7$ k	0.61 $\pm.02$	0.56 $\pm.03$	0.55 $\pm.02$	0.50 $\pm.02$	0.46 $\pm.03$	0.45 $\pm.03$	0.40 $\pm.03$	0.36 $\pm.04$	0.36 $\pm.03$
Age $>70$	${\sim}6$ k	0.59 $\pm.03$	0.55 $\pm.04$	0.52 $\pm.02$	0.47 $\pm.03$	0.43 $\pm.04$	0.41 $\pm.03$	0.37 $\pm.04$	0.34 $\pm.05$	0.32 $\pm.03$

Appendix

Table 9: Demographic subgroup analysis on MIMIC-IV-ECG test set. Macro-F1 (mean

\pm

std, 5 seeds) for MINN, ResNet+FiLM (multimodal, no manifold; metadata-matched reference), and the strongest unimodal baseline per electrolyte.

Subgroup	$n$	ECGFounder	Logit Adj.	Manifold Mixup	ResNet+FiLM	Poisson Learn.	MINN
K-sparing diuretics + ACEi	142	0.40 $\pm.05$	0.46 $\pm.06$	0.43 $\pm.05$	0.49 $\pm.05$	0.47 $\pm.05$	0.53 $\pm.05$
Age $\geq$ 85 + $\geq$ 8 medications	238	0.45 $\pm.03$	0.46 $\pm.03$	0.45 $\pm.04$	0.47 $\pm.04$	0.46 $\pm.03$	0.50 $\pm.03$
Dialysis patients	487	0.38 $\pm.02$	0.42 $\pm.03$	0.41 $\pm.03$	0.46 $\pm.03$	0.44 $\pm.02$	0.51 $\pm.02$
Heart failure + concurrent hypoK	195	0.44 $\pm.04$	0.46 $\pm.04$	0.46 $\pm.04$	0.48 $\pm.04$	0.47 $\pm.04$	0.52 $\pm.03$

Appendix

Table 10: Subgroup stress test: K + Macro-F1 for underrepresented patient subgroups (mean

\pm

std, 5 seeds). Best in bold . Each baseline column tests a distinct failure mode (see text). The cohort ‘Heart failure + concurrent hypoK’ comprises heart-failure patients with

\geq 1

prior hypoK lab draw in the EHR; the test-draw label distribution remains 3-class.

Setting	Method	K + F1 ( $\uparrow$ )	Ca 2+ F1 ( $\uparrow$ )	Mg 2+ F1 ( $\uparrow$ )
In-distribution	ECGFounder	0.55 $\pm.01$	0.44 $\pm.03$	0.35 $\pm.03$
In-distribution	MINN	0.61 $\pm.02$	0.50 $\pm.02$	0.40 $\pm.03$
Temporal split (2008–2016 $\to$ 2017–2019)	ECGFounder	0.51 $\pm.03$	0.39 $\pm.04$	0.30 $\pm.05$
Temporal split (2008–2016 $\to$ 2017–2019)	MINN	0.59 $\pm.04$	0.48 $\pm.04$	0.39 $\pm.05$
Care-unit split (ICU $\to$ ED)	ECGFounder	0.47 $\pm.05$	0.34 $\pm.06$	0.27 $\pm.06$
Care-unit split (ICU $\to$ ED)	MINN	0.57 $\pm.04$	0.46 $\pm.05$	0.37 $\pm.05$

Appendix

Table 11: Within-MIMIC distribution-shift stress tests: K + /Ca 2+ /Mg 2+ Macro-F1 (mean

\pm

std, 5 seeds). In-distribution rows repeat Table 1 headlines for reference.

Component	Hyperparameter	Value
Encoder (ResNet-18)	Input channels	12
	Embedding dim $d$	128
	Dropout	0.3
FiLM Fusion	Metadata dim $p$	40
FiLM Fusion	Output dim	128
Source Network $s_{\phi}$	Hidden layers	3

Appendix

Table 12: Hyperparameter settings for MINN.

Method	K + ( $\uparrow$ )	Ca 2+ ( $\uparrow$ )	Mg 2+ ( $\uparrow$ )
S4/S4D-ECG	0.52 $\pm.03$	0.41 $\pm.03$	0.33 $\pm.04$
LDAM+DRW	0.53 $\pm.02$	0.43 $\pm.02$	0.34 $\pm.03$
Focal Loss	0.51 $\pm.03$	0.42 $\pm.03$	0.33 $\pm.03$
SMOTE	0.49 $\pm.03$	0.40 $\pm.03$	0.32 $\pm.04$
Label Propagation	0.46 $\pm.04$	0.37 $\pm.04$	0.30 $\pm.04$
MINN	0.61 $\pm.02$	0.50 $\pm.02$	0.40 $\pm.03$

Appendix

Table 13: Appendix baseline results (Macro-F1, 100% labeled data, mean

\pm

std, 5 seeds).

	K +		Ca 2+		Mg 2+
Method	Hypo ( $\uparrow$ )	Hyper ( $\uparrow$ )	Hypo ( $\uparrow$ )	Hyper ( $\uparrow$ )	Hypo ( $\uparrow$ )	Hyper ( $\uparrow$ )
ECGFounder	0.38 $\pm.02$	0.34 $\pm.03$	0.28 $\pm.03$	0.22 $\pm.03$	0.20 $\pm.03$	0.18 $\pm.04$
Poisson Learning	0.37 $\pm.03$	0.33 $\pm.03$	0.27 $\pm.03$	0.21 $\pm.03$	0.19 $\pm.04$	0.17 $\pm.03$
Logit Adj.	0.36 $\pm.02$	0.31 $\pm.03$	0.26 $\pm.03$	0.20 $\pm.03$	0.18 $\pm.03$	0.16 $\pm.03$
MINN	0.45 $\pm.02$	0.40 $\pm.02$	0.35 $\pm.02$	0.28 $\pm.02$	0.26 $\pm.03$	0.22 $\pm.03$

Appendix

Table 14: Per-class AUCPR for hypo and hyper classes (mean

\pm

std, 5 seeds). Best in bold .

Method	ECE ( $\downarrow$ )
ECGFounder	0.124 $\pm.012$
Poisson Learning	0.116 $\pm.011$
Logit Adj.	0.105 $\pm.010$
MINN	0.083 $\pm.009$

Appendix

Table 15: ECE (15 bins) on K + test set (mean

\pm

std, 5 seeds).

		Labeled fraction
Electrolyte	Method	1%	5%	10%	25%	50%	100%
K +	ECGFounder	0.29 $\pm.04$	0.37 $\pm.03$	0.43 $\pm.03$	0.49 $\pm.02$	0.53 $\pm.02$	0.55 $\pm.01$
	Poisson Learning	0.27 $\pm.04$	0.35 $\pm.03$	0.40 $\pm.03$	0.44 $\pm.02$	0.47 $\pm.02$	0.49 $\pm.03$
	Logit Adj.	0.25 $\pm.04$	0.33 $\pm.03$	0.41 $\pm.03$	0.47 $\pm.02$	0.51 $\pm.02$	0.53 $\pm.02$
	MINN	0.38 $\pm.05$	0.47 $\pm.03$	0.52 $\pm.02$	0.56 $\pm.02$	0.59 $\pm.02$	0.61 $\pm.02$
Ca 2+	ECGFounder	0.28 $\pm.04$	0.33 $\pm.03$	0.36 $\pm.03$	0.40 $\pm.02$	0.42 $\pm.02$	0.44 $\pm.03$

Appendix

Table 16: Macro-F1 at reduced labeled fractions for all three electrolytes (mean

\pm

std, 5 seeds). Best in bold .

Comparison	Electrolyte	Mean $\Delta$ F1 (MINN $-$ baseline)	Paired $t$ -test $p$
MINN vs. ECGFounder	K +	$+0.06$	$0.003$
MINN vs. ECGFounder	Ca 2+	$+0.06$	$0.012$
MINN vs. Logit Adj.	K +	$+0.08$	$0.001$
MINN vs. Logit Adj.	Ca 2+	$+0.07$	$0.004$

Appendix

Table 17: Paired

t

-test

p

-values for headline comparisons (5 seeds, two-sided). MINN vs. ECGFounder for K + and Ca 2+ both reject the null at

p<0.05

; MINN vs. Logit Adj. shown for reference.

Asset	Type	Version	License
MIMIC-IV [ 12 ]	Dataset	2.2	PhysioNet credentialed (DUA)
MIMIC-IV-ECG [ 38 ]	Dataset	0.3	PhysioNet credentialed (DUA)
ECGFounder [ 6 ]	Model weights	v1.0	per upstream release
ECG-FM [ 7 ]	Model weights	v0.1	per upstream release
TabPFN [ 34 ]	Library	0.1.x	Apache-2.0
CatBoost [ 35 ]	Library	1.2	Apache-2.0

Appendix

Table 18: Third-party assets used in this work. Versions reflect the snapshot used for reported experiments; licenses are the upstream license at the time of access.

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