# Informed Machine Learning with Knowledge Landmarks

arXiv: https://arxiv.org/abs/2604.00256 · HTML: https://mlsanity.com/paper/2604.00256/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 45 references that are in the map.
- First posted on arXiv 31 Mar 2026.

## Builds on (most cited first, 17 of 17)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [A Survey on Self-supervised Learning: Algorithms, Applications, and Future Trends](https://mlsanity.com/paper/2301.05712/) (2023)
- [Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems](https://mlsanity.com/paper/1903.12394/) (2019)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems](https://mlsanity.com/paper/2003.04919/) (2020)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
- [A Survey on Deep Active Learning: Recent Advances and New Frontiers](https://mlsanity.com/paper/2405.00334/) (2024)
- [SoftAdapt: Techniques for Adaptive Loss Weighting of Neural Networks with Multi-Part Loss Functions](https://mlsanity.com/paper/1912.12355/) (2019)
- [Analyzing Differentiable Fuzzy Implications](https://mlsanity.com/paper/2006.03472/) (2020)
- [Domain knowledge-guided machine learning framework for state of health estimation in Lithium-ion batteries](https://mlsanity.com/paper/2409.14575/) (2024)
- [S2FS: Spatially-Aware Separability-Driven Feature Selection in Fuzzy Decision Systems](https://mlsanity.com/paper/2509.25841/) (2025)

## Abstract (arXiv)

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.