Informed Machine Learning with Knowledge Landmarks

Chuyi Dai; Witold Pedrycz; Suping Xu; Ding Liu; Xianmin Wang

cs.LGMar 31, 2026

Informed Machine Learning with Knowledge Landmarks

Authors: Chuyi Dai, Witold Pedrycz, Suping Xu, Ding Liu, Xianmin Wang

Organizations: Department of Electrical and Computer Engineering, University of Alberta, Edmonton, T6G 1H9, Canada · School of ElectroMechanical Engineering, Xidian University, Xi’an 710071, China · Hubei Subsurface Multi-Scale Imaging Key Laboratory, Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan 430074, China

Abstract

Informed Machine Learning has emerged as a viable generalization of Machine Learning (ML) by building a unified conceptual and algorithmic setting for constructing models on a unified basis of knowledge and data. Physics-informed ML involving physics equations is one of the developments within Informed Machine Learning. This study proposes a novel direction of Knowledge-Data ML, referred to as KD-ML, where numeric data are integrated with knowledge tidbits expressed in the form of granular knowledge landmarks. We advocate that data and knowledge are complementary in several fundamental ways: data are precise (numeric) and local, usually confined to some region of the input space, while knowledge is global and formulated at a higher level of abstraction. The knowledge can be represented as information granules and organized as a collection of input-output information granules called knowledge landmarks. In virtue of this evident complementarity, we develop a comprehensive design process of the KD-ML model and formulate an original augmented loss function L, which additively embraces the component responsible for optimizing the model based on available numeric data, while the second component, playing the role of a granular regularizer, so that it adheres to the granular constraints (knowledge landmarks). We show the role of the hyperparameter positioned in the loss function, which balances the contribution and guiding role of data and knowledge, and point to some essential tendencies associated with the quality of data (noise level) and the level of granularity of the knowledge landmarks. Experiments on two physics-governed benchmarks demonstrate that the proposed KD model consistently outperforms data-driven ML models.

Figures & tables

Fig. 1: Input space $\Omega$ with a collection of knowledge landmarks $(A_{i},B_{i})$ and limited space $\Omega^{\ast}$ for which numeric data are generated by $f(\boldsymbol{x};\boldsymbol{W})$ for some specific numeric values of $\boldsymbol{W}$ , namely $\boldsymbol{w}_{0}$ .

Fig. 2: Matching of an information granule $B$ and a numeric entity $x_{0}$ .

Variable	Symbol	Range	Unit
Input Variables
Spatial Coordinate	$s$	$[0,3]$	m
Time	$t$	$(0,60]$	s
Parameters
Spilled Mass	$R$	$[7,13]$	kg
Diffusion Rate	$Y$	$[0.02,0.12]$	m 2 /s

TABLE I: Variable Ranges and Parameters for Environmental Pollutant Dispersion

Panel of Figure 4 — Fig. 3: Spatiotemporal domain and local observation setup for the environmental benchmark. (a) Pollutant concentration field $f(s,t)$ over $\Omega$ , computed with baseline parameters. (b) Domain $\Omega$ and local observation window $\Omega_{1}^{\ast}$ in the $(s,t)$ space.

Fig. 4: Output space granulation for the environmental model. The histogram shows the distribution of scaled concentration values in $\Omega$ , and curves represent five Gaussian membership functions defining the output contexts.

Fig. 5: Input granules $\{A_{ik}\}$ for the environmental model, obtained via conditional FCM for each output context. Each subplot shows eight input granules associated with one output context $B_{i}$ , where the color of each point represents its membership degree.

Fig. 6: Curves of $Q_{2}(\lambda)$ , and $Q_{1}(\lambda)+Q_{2}(\lambda)$ for the environmental benchmark as functions of the trade-off parameter $\lambda$ .

Position of $\Omega^{\ast}$	$\lambda_{\mathrm{opt}}$	KD -model			Data-based model ( $\lambda=1$ )			$Q_{1}+Q_{2}$ Improvement
Position of $\Omega^{\ast}$	$\lambda_{\mathrm{opt}}$	$Q_{1}$	$Q_{2}$	$Q_{1}+Q_{2}$	$Q_{1}$	$Q_{2}$	$Q_{1}+Q_{2}$	$Q_{1}+Q_{2}$ Improvement
$\Omega_{1}^{\ast}$	0.88	0.012	6.626	6.638	0.002	7.816	7.819	15.10%
$\Omega_{2}^{\ast}$	0.98	0.009	6.613	6.622	0.004	7.163	7.167	7.60%
$\Omega_{3}^{\ast}$	0.32	0.095	6.672	6.767	0.794	7.883	8.678	22.02%
$\Omega_{4}^{\ast}$	0.74	0.010	6.949	6.959	0.010	7.420	7.431	6.34%

TABLE II: Improvement of the total objective

Q=Q_{1}+Q_{2}

for Environmental Pollutant Dispersion.

Fig. 7: Different positions of $\Omega^{\ast}$ for the environmental benchmark: $\Omega_{2}^{\ast}$ , $\Omega_{3}^{\ast}$ , and $\Omega_{4}^{\ast}$ .

Variable	Symbol	Range	Unit
Input Design Variables
Piston Weight	$\xi$	$[30,60]$	kg
Surface Area	$\Gamma$	$[0.005,0.020]$	m 2
Parameters
Gas Volume	$V_{0}$	$[0.002,0.010]$	m 3
Spring Coeff.	$k$	$[1000,5000]$	N/m

TABLE III: Variable Ranges and Parameters for Piston Simulation

Panel of Figure 11 — Fig. 8: Spatiotemporal domain and local observation setup for the piston benchmark. (a) Cycle-time response surface $\psi(\xi,\Gamma)$ over the design space. (b) Full domain $\Omega$ and local subdomain location $\Omega_{1}^{\ast}$ for the piston model.

Fig. 9: Output space granulation for the Piston Cycle-Time Simulation benchmark.

Fig. 10: Input granules $\{A_{ik}\}$ for the Piston Cycle-Time Simulation benchmark. Each subplot shows five input granules associated with one output context $B_{i}$ , where the color of each point represents its membership degree.

Fig. 11: Curves of $Q_{2}(\lambda)$ , and $Q_{1}(\lambda)+Q_{2}(\lambda)$ for the piston benchmark as functions of the trade-off parameter $\lambda$ .

Position of $\Omega^{\ast}$	$\lambda_{\mathrm{opt}}$	KD -model			Data-based model ( $\lambda=1$ )			$Q_{1}+Q_{2}$ Improvement
Position of $\Omega^{\ast}$	$\lambda_{\mathrm{opt}}$	$Q_{1}$	$Q_{2}$	$Q_{1}+Q_{2}$	$Q_{1}$	$Q_{2}$	$Q_{1}+Q_{2}$	$Q_{1}+Q_{2}$ Improvement
$\Omega_{1}^{\ast}$	0.90	0.0001	0.0968	0.0977	0.0011	0.2540	0.2551	61.70%
$\Omega_{2}^{\ast}$	0.78	0.0002	0.0921	0.0923	0.0004	0.1359	0.1363	32.27%
$\Omega_{3}^{\ast}$	0.36	0.0004	0.0912	0.0916	0.0007	0.1345	0.1352	32.28%
$\Omega_{4}^{\ast}$	0.10	0.0004	0.0927	0.0931	0.0009	0.2279	0.2288	59.32%

TABLE IV: Improvement of the total objective

Q=Q_{1}+Q_{2}

for Piston Cycle-Time Simulation.

Fig. 12: Different positions of $\Omega^{\ast}$ for the piston benchmark: $\Omega_{2}^{\ast}$ , $\Omega_{3}^{\ast}$ , and $\Omega_{4}^{\ast}$ .

Panel of Figure 17 — Fig. 13: Selected $\lambda_{\mathrm{opt}}$ versus noise level. Points denote repeated runs; diamonds denote medians. (a) Environmental benchmark. (b) Piston benchmark.

Fig. 14: Sensitivity of the trade-off parameter $\lambda_{\mathrm{opt}}$ to the parameter-space width ratio $r$ for both benchmarks. The dashed line corresponds to the Environmental Pollutant Dispersion benchmark and the solid line to the Piston Cycle-Time Simulation benchmark.

Scientific ML

Scientific machine learning combines data-driven models with scientific knowledge to analyze, predict, and simulate complex systems.

See all 122 papers

A PAC-Bayesian View of Generalisation for Physics-Informed Machine Learning
May 25, 2026Thien V. Nguyen, Amaury Habrard, Benjamin GuedjPAC-Bayesian Generalization Bounds Physics-Informed ML
Université Jean Monnet Saint-Étienne, CNRS, Institut d’Optique Graduate School, Laboratoire Hubert Curien UMR 5516, F-42023, Saint-Étienne, France · Université Jean Monnet Saint-Étienne, CNRS, Institut d’Optique Graduate School, Laboratoire Hubert Curien UMR 5516, Inria, F-42023, Saint-Étienne, France · Institut Universitaire de France +1
PIKS: Universal Physics-Informed Kernel Methods
Jul 29, 2026Joachim Bona-Pellissier, Giacomo Meanti, Matteo Santacesaria +1Kernel Methods Physics-Informed ML
MaLGa Center, DIBRIS, Università degli Studi di Genova, Genoa, Italy · MaLGa Center, DIMA, Università degli Studi di Genova, Genoa, Italy · Istituto Italiano di Tecnologia, Genoa, Italy
Automatically Building and Updating a Knowledge Graph of MLIP Models
Oct 7, 2026Alexis Beer, Liudmyla Klochko, Mathieu d'AquinKG Construction
MosAIk Team, LORIA, Université de Lorraine, CNRS, Nancy, France · CRET AIREL, Université de Lorraine, Nancy, France

May 25, 2026J/K move · Enter open · S save

A PAC-Bayesian View of Generalisation for Physics-Informed Machine Learning

Thien V. Nguyen, Amaury Habrard, Benjamin Guedj

Université Jean Monnet Saint-Étienne, CNRS, Institut d’Optique Graduate School, Laboratoire Hubert Curien UMR 5516, F-42023, Saint-Étienne, France · Université Jean Monnet Saint-Étienne, CNRS, Institut d’Optique Graduate School, Laboratoire Hubert Curien UMR 5516, Inria, F-42023, Saint-Étienne, France · Institut Universitaire de France+1

PAC-Bayesian Generalization Bounds Physics-Informed ML Statistical Learning Theory Generalization Bounds

Open paper AskPDF arXiv

Physics-informed machine learning (PIML) integrates mechanistic knowledge, typically in the form of partial differential equations (PDE), into data-driven models. Despite strong empirical performance, its statistical generalisation properties remain poorly understood, particularly in the regression setting with unbounded losses. Existing analyses rely on approximation or stability arguments and do not fully capture how physical structure influences generalisation from finite data. In this work, we develop a PAC-Bayesian framework for PIML that provides high-probability generalisation guarantees in the presence of unbounded losses. We adopt a multi-task perspective that jointly treats data fidelity, PDE residuals, initial and boundary conditions, avoiding the looseness induced by standard union-bound approaches. Our analysis leverages the structure of physics-informed objectives to derive novel bounds where the complexity scales with input-gradient norms of the losses, revealing a direct link between physical regularity and generalisation. We instantiate this framework under Sobolev and Poincaré-type assumptions, yielding two classes of bounds that trade off statistical complexity and smoothness in different regimes. Building on these results, we propose a self-bounding-aware learning algorithm that directly optimises tractable surrogates of the derived bounds, along with a practical procedure to estimate the associated constants in realistic settings. Empirical evaluations on standard PDE benchmarks demonstrate that our bounds are non-vacuous, significantly tighter than union-bound baselines, and can be effectively minimised during training. Overall, our results provide a principled statistical foundation for the generalisation of physics-informed models.

Abstract

Scientific ML

Explore similar work

A PAC-Bayesian View of Generalisation for Physics-Informed Machine Learning

PIKS: Universal Physics-Informed Kernel Methods

Automatically Building and Updating a Knowledge Graph of MLIP Models