Informed machine learning

Overview

Informed machine learning means learning from data together with prior knowledge that exists apart from the learning task and is written in a formal form 1903.12394 §2.2 1903.12394 §2.2.2. The knowledge can be a physical equation, a logic rule, the output of a simulator, an invariance, a knowledge graph, a probability relation or a person's feedback 1903.12394 §4.2.

The motivation is that data alone can fall short when data is scarce, when a model must obey natural laws or safety rules, or when a result has to be explainable 1903.12394 §1. The survey gives one shared vocabulary for these methods, the taxonomy below, which covers the whole pipeline rather than one technique 1903.12394 §2 (card: Informed Machine Learning: A Taxonomy and Survey).

Related names for overlapping work include physics-informed deep learning, physics-guided neural networks and semantic-based regularisation 1903.12394 §1. See physics-informed neural networks for the most common physics case.

Taxonomy

Each method can be placed on three axes 1903.12394 §4:

1. Knowledge source. Scientific, world or expert knowledge. The list is "neither complete nor disjoint", so one source may fall into several categories 1903.12394 §4.1.
2. Knowledge representation. Eight forms, from equations and simulation results to logic rules, graphs, symmetries, probability assumptions and human input 1903.12394 §4.2.
3. Integration point. Where the knowledge enters the pipeline: the training data, the hypothesis set, the learning algorithm, or the final hypothesis 1903.12394 §4.3.

Variants

Variants are grouped by integration point. A method can use more than one.

Through the training data

Knowledge is turned into extra labelled or unlabelled samples, most often simulation results 1903.12394 §3.2.3. The survey lists this as the usual route when the goal is to need less real data 1903.12394 §3.2.3. It also says that data augmentation with virtual examples is mathematically equivalent to a regulariser, citing the source for this claim without deriving it 1903.12394 §5.4. Simulated data can differ from real data, and the survey names adversarial domain adaptation and domain randomisation as directions, without measured effects 1903.12394 §5.3 1903.12394 §7.

Through the hypothesis set (architecture)

The structure of the model encodes the knowledge. Knowledge graphs, spatial invariances and logic rules often enter here 1903.12394 §3.2.3. The clearest example is KBANN, which builds a network from propositional rules and then refines its weights by learning 1903.12394 §5.5. The survey lists whether this approach works for deep networks as an open question 1903.12394 §7.

Through the learning algorithm (loss)

Knowledge becomes an extra term in the training objective. Equations, logic rules, probability relations and human feedback most often change the algorithm this way 1903.12394 §3.2.3. The example in the text is a knowledge term that penalises violations of the given equations, with weights on the label, regulariser and knowledge terms 1903.12394 eq. 1 1903.12394 §5.1. The knowledge term needs only inputs and the model, not labels 1903.12394 §5.1. The loss term is the form used by physics-informed neural networks.

Logic rules must become differentiable to be used in this way. The survey mentions relaxing them "for example via the t-norm" but does not pick one 1903.12394 §5.5.

Through the final hypothesis

The model's output is checked or corrected against the knowledge. The survey classes this as aiming at knowledge conformity 1903.12394 §3.2.3. The survey gives no method details for this route beyond the classification.

When to use which

The survey does not benchmark the methods. The table maps a constraint to the integration point that the surveyed literature uses for it. It does not rank the options.

Note that most papers in the survey aim at better performance 1903.12394 §3.2.3.

Implementation sketch

Common core: a single objective that adds a knowledge term to the usual label loss and regulariser 1903.12394 eq. 1. The variants differ in what the knowledge term measures, and where the knowledge enters if it is not a loss term.

# Objective with a knowledge term  [arxiv:1903.12394 eq. 1] [§5.1]
# Input:  model f, labelled pairs (x_i, y_i), weights lam_l, lam_r, lam_k
# Output: scalar objective to minimise

def informed_objective(f, xs, ys, lam_l, lam_r, lam_k):
    # Label term: usual loss over the labelled samples  [eq. 1]
    label_term = sum(loss(f(x), y) for x, y in zip(xs, ys))
    # Regulariser R(f) on the model  [eq. 1]
    reg_term = regulariser(f)
    # Knowledge term: penalises violations of the given equations  [§5.1]
    knowledge_term = knowledge_loss(f, xs)
    return lam_l * label_term + lam_r * reg_term + lam_k * knowledge_term

The survey does not set the form of knowledge_loss or the values of the weights. It names the weighting as a challenge 1903.12394 §7.

Contradictions

None known. The survey reports no experiments, so no numbers conflict. Names for overlapping methods vary 1903.12394 §1, which can make results hard to compare.

Open problems

• Weights. How to balance knowledge against labels is open 1903.12394 §7.
• Rule learning. How to learn rules when none are given is open 1903.12394 §7.
• Evaluation. Knowledge graphs are often customised per paper, so comparing them is hard 1903.12394 §7.
• Coverage. The survey does not list its sources, search terms or date range, and says it does not claim to cover all research 1903.12394 §1 1903.12394 §3.1.2. Treat its "main paths" as a sample, not a census.

Papers

• Informed Machine Learning: A Taxonomy and Survey of Integrating Knowledge into Learning Systems: the taxonomy (source, representation, integration point) and a survey of how prior knowledge is integrated.