Causal-PIK: Physics-Informed Causal Reasoning

• Causal-PIK is a methodology that integrates causal modeling with a physics-informed kernel to predict the effects of actions in dynamic physical environments.
• Its innovative kernel replaces standard similarity metrics with measures capturing directional and magnitude similarities of predicted state changes.
• By leveraging Bayesian optimization guided by causal predictions, Causal-PIK achieves superior sample efficiency and performance on challenging physical reasoning benchmarks.

Causal-PIK refers to “Causality-based Physical Reasoning with a Physics-Informed Kernel,” a methodology developed to solve complex physical reasoning tasks by integrating causal modeling, model-based prediction, and Bayesian optimization using a tailored, physically informed kernel (Parés-Morlans et al., 28 May 2025). The approach is designed for settings in which agents must reason about the causal effects of actions in environments governed by unknown or intricate dynamics (as typified by benchmark tasks like Virtual Tools and PHYRE). By explicitly predicting the physical consequences of candidate actions and leveraging these predictions to guide efficient exploration, Causal-PIK demonstrates superior sample efficiency and performance compared to both traditional machine learning baselines and human problem solvers on challenging physical reasoning tasks.

1. Causal-PIK Methodology: Integration of Causality and Physical Intuition

Causal-PIK centers its search for solutions around the principle that actions exert causal effects on a physical scene. Rather than relying solely on model-free reinforcement learning or random exploration, it first learns a forward dynamics model that predicts the state change of relevant objects resulting from an action given the current environment. For every candidate action, this model estimates the immediate causal effect on each dynamic object after a key “event” (such as first contact or motion), computing the resulting change in state as

$$\dot{s}^O = (\mathbf{s}^O_{t_\mathrm{event}+\Delta t} - \mathbf{s}^O_{t_\mathrm{event}})/\Delta t,$$

where $\mathbf{s}^O_{t}$ denotes the state vector for object $O$ at time $t$. This predicted effect forms the basis for subsequent clustering and exploration.

By grouping actions whose predicted consequences are causally similar—i.e., those giving rise to similar object motions—Causal-PIK seeks to construct a more informative and sample-efficient search process, enabling the agent to focus on promising regions of the action space.

2. Physics-Informed Kernel: Construction and Mathematical Formulation

The innovation of Causal-PIK lies in its “Physics-Informed Kernel” (or PIK), which replaces standard input-space similarity (such as the squared Euclidean distance used by RBF kernels) with a similarity measure over predicted physical effects. For two candidate actions $a$ and $b$, the causal similarity for a given object $O$ is computed as:

• Directional similarity: Cosine similarity of state change vectors,

$$\textrm{sim}_\textrm{cos}(\dot{s}^{O,a}, \dot{s}^{O,b}) = \frac{\dot{s}^{O,a} \cdot \dot{s}^{O,b}}{ \|\dot{s}^{O,a}\| \cdot \|\dot{s}^{O,b}\| }$$

• Magnitude similarity:

$$\textrm{sim}_\textrm{mag}(\dot{s}^{O,a}, \dot{s}^{O,b}) = \frac{1}{1 + |\|\dot{s}^{O,a}\| - \|\dot{s}^{O,b}\||}$$

• The per-object similarity, $\textrm{sim}_\textrm{obj}(O, a, b)$, is the maximum of the two, rectified at zero.

These are then aggregated over all $D$ dynamic objects into the final causal similarity:

$$\textrm{sim}_\textrm{csl}(a, b) = \frac{1}{D}\sum_{O=1}^{D} \textrm{sim}_\textrm{obj}(O, a, b) \cdot \exp\left(\left(\frac{1}{D}\sum_{O=1}^{D} \textrm{sim}_\textrm{obj}(O, a, b)\right)-1\right)$$

This similarity function is employed as the kernel in a Gaussian process model, guaranteeing symmetry and positive semi-definiteness.

3. Bayesian Optimization and Efficient Search

The physics-informed kernel enables the integration of Bayesian optimization into the causal reasoning process. The agent maintains a Gaussian process posterior over the success metric $f(x)$ (e.g., distance to goal) as a function of action parameter $x$, using the physics-informed kernel to encode prior knowledge about which actions are causally similar. At each round, the algorithm:

• Generates a candidate pool (e.g., via a Sobol sequence);
• Scores candidates using an acquisition function (Upper Confidence Bound, UCB);
• Simulates the most promising actions in an imaginative, low-fidelity dynamics model to further refine estimates;
• Executes the highest-ranked action in the real environment.

This structure allows both exploitation of known promising actions and exploration of new, causally distinct behaviors, with the clustering induced by the kernel providing a mechanism for rapid generalization across the action space.

4. Experimental Results: Virtual Tools and PHYRE Benchmarks

In empirical evaluation, Causal-PIK outperforms both baseline and state-of-the-art sample-efficient solvers on the Virtual Tools and PHYRE benchmarks. On Virtual Tools, with a limit of 10 attempts per task, Causal-PIK achieves an AUCCESS of 65.0 (±25.0), exceeding that of SSUP (58.0±27.0) and human baselines (53.25±23). In the highly challenging PHYRE benchmark with a full hybrid action space of over 2.5 million possible actions, Causal-PIK delivers an AUCCESS over 10 points higher than the best competitors operating without action space reduction.

For both benchmarks, the method demonstrates a rapid convergence toward successful solutions, reducing the number of interactive trials required compared to methods based on random search (RAND), DQN, or less informed Bayesian optimization.

5. Human User Study and Alignment with Physical Intuition

A systematic user paper was conducted on PHYRE tasks by recruiting 50 human participants, each attempting up to 10 trials per puzzle. Human AUCCESS scores were found to be competitive but consistently below those of Causal-PIK, with mean scores around 36.6 (±10.0) compared to the method’s 41.6 (±9.33). Importantly, per-puzzle performance showed a strong correlation ($r=0.73$) between human and Causal-PIK scores, indicating that the system’s causal reasoning process parallels human physical intuition. In particularly challenging tasks (e.g., those rarely solved by humans within the time limit), Causal-PIK often succeeded by virtue of its more systematic causal exploration and tighter clustering of promising actions.

6. Applications and Broader Significance

Causal-PIK’s approach—reasoning about the future causal effects of candidate actions and using physical dynamics to structure exploration—is applicable to scenarios where action efficiency is crucial and direct experimentation is costly:

• Robotics: Reducing the number of physical trials required for manipulation or navigation tasks.
• Simulation-based design: Efficiently identifying optimal interventions in computationally intensive environments.
• Autonomous systems: Dealing with real-time constraints and complex environments with unknown physics.

By incorporating causal predictions into the Bayesian optimization process, Causal-PIK advances the frontier in sample-efficient decision making under physical uncertainty, demonstrating practical relevance for both artificial agents and the understanding of human-like physical reasoning strategies.

7. Limitations and Future Directions

Potential limitations of Causal-PIK include dependence on the accuracy of the learned dynamics model and the cost of initial model training. Scalability to increasingly complex or high-noise environments may require further engineering of the forward model or kernel design. Future research may explore:

• Extensions to multi-step or long-term causal reasoning.
• Integration with other forms of prior knowledge (semantic or symbolic).
• Real-world deployment in robotic or industrial systems.

These prospects suggest Causal-PIK establishes a methodological template for embedding physically grounded causal reasoning at the heart of adaptive, efficient action in complex environments.