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.

Electrocardiogram Classification

Electrocardiogram classification uses machine learning to identify heart rhythms and cardiac conditions from ECG waveforms for screening, diagnosis, and monitoring.

See all 108 papers

Physics-Informed Implicit Neural Representations for Improved Myocardial Perfusion MRI Quantification
Aug 11, 2026Christos Tsepas, Chang Yan, Maximilian Fuetterer +2Implicit Neural Representations Medical Image Analysis
Department of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, the Netherlands · Institute for Biomedical Engineering, University and ETH Zurich, Zurich, Switzerland
ECG for laboratory values:A comprehensive physiological and technical analysis
Oct 20, 2026Walentin PalanchukClinical Prediction Electrocardiography
Carl von Ossietzky Universität Oldenburg Fach-Bachelor Informatik
Physiology and Anatomy Aware Inverse Inference of Myocardial Infarction for Cardiac Digital Twin
May 21, 2026Mengxiao Wang, Yilin Lyu, Julia Camps +6Electrocardiography 3D Medical Image Segmentation
Shanghai Jiao Tong University, Shanghai, China · National University of Singapore, Singapore · Universitat Pompeu Fabra, Barcelona, Spain

Aug 11, 2026J/K move · Enter open · S save

Physics-Informed Implicit Neural Representations for Improved Myocardial Perfusion MRI Quantification

Christos Tsepas, Chang Yan, Maximilian Fuetterer, Sebastian Kozerke+1

Department of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, the Netherlands · Institute for Biomedical Engineering, University and ETH Zurich, Zurich, Switzerland

Implicit Neural Representations Medical Image Analysis Physics-Informed ML Cardiac MRI

Open paper AskPDF arXiv

Quantifying myocardial perfusion from cardiac magnetic resonance (CMR) can be achieved by fitting tracer-kinetic models to the dynamic contrast-enhanced MR data. However, fitting the observed data with multi-compartment exchange models, which describe the evolution of the contrast agent in the tissue, to estimate perfusion parameters is a challenging inverse problem that is sensitive to noise and acquisition variability. Previously, physics-informed neural networks (PINNs) have been proposed as an alternative to conventional non-linear least squares fitting methods with promising results for quantitative perfusion CMR. In this work, we extend the previously proposed PINN framework with spatiotemporal implicit neural representations (INRs) to represent the MR signal as a continuous spatiotemporal function and to improve the accuracy, smoothness, and physical consistency of the PINN model. In realistic simulated CMR datasets, our proposed PINN with INRs demonstrates improved robustness and parameter estimation accuracy over the previously established methods. The code is available at https://github.com/q-cardIA/pinn-inr.

Abstract

Electrocardiogram Classification

Explore similar work

Physics-Informed Implicit Neural Representations for Improved Myocardial Perfusion MRI Quantification

ECG for laboratory values:A comprehensive physiological and technical analysis

Physiology and Anatomy Aware Inverse Inference of Myocardial Infarction for Cardiac Digital Twin