Causal-PIK: Causality-based Physical Reasoning with a Physics-Informed Kernel
Authors: Carlota Parés Morlans
Michelle Yi
Claire Chen
Sarah A Wu
Rika Antonova
Tobias Gerstenberg
Jeannette Bohg
Venue: ICML 2025 poster (2025)
Keywords: active exploration
physical reasoning
causality
bayesian optimization
Primary area: general_machine_learning->causality
TL;DR: We propose a method that leverages Bayesian optimization to reason about causality via a Physics-Informed Kernel. This helps agents to efficiently find solutions to tasks that involve complex interactions between objects with unknown dynamics.

Abstract: Tasks that involve complex interactions between objects with unknown dynamics make planning before execution difficult. These tasks require agents to iteratively improve their actions after actively exploring causes and effects in the environment. For these type of tasks, we propose Causal-PIK, a method that leverages Bayesian optimization to reason about causal interactions via a Physics-Informed Kernel to help guide efficient search for the best next action. Experimental results on Virtual Tools and PHYRE physical reasoning benchmarks show that Causal-PIK outperforms state-of-the-art results, requiring fewer actions to reach the goal. We also compare Causal-PIK to human studies, including results from a new user study we conducted on the PHYRE benchmark. We find that Causal-PIK remains competitive on tasks that are very challenging, even for human problem-solvers.

Causal-PIK:
Causality- Physical Reasoning with a Physics-Informed Kernel
Carlota Parés-Morlans Michelle Yi!
Claire Chen
Sarah A. Wu Rika Antonova 13 Tobias Gerstenberg'
Jeannette Bohg'
Abstract
2023). These models encode our causal understanding of the domain - our beliefs about how one event leads to another.
Tasks that involve complex interactions between objects with unknown dynamics make planning before execution difficult. These tasks require agents to iteratively improve their actions after actively exploring causes and effects in the en- vironment. For these type of tasks, we propose Causal-PIK, a method that leverages Bayesian optimization to reason about causal interactions via Physics-Informed Kernel help guide effi- cient search for the best next action. Experimen- tal results on Virtual Tools and PHYRE physical reasoning benchmarks show that Causal-PIK out- performs State-of-the-ae results, requiring fewer actions to reach the goal. We also compare Causal- PIK to human studies, including results from a new user study we conducted on the PHYRE benchmark. We find that Causal-PIK remains competitive on tasks that are very challenging. even for human problem-solvers.
While predicting the exact sequence of events is difficult, our physical intuition allows us to estimate the immediate causal effects of our actions, such as which object will move and in which direction. In addition, we rapidly learn from the outcomes of previous actions (Allen et al., 2020). If the red ball almost landed in the goal region, we would slightly change our action and try again. If was way off, we might try something completely different. Inspired by people's s rapid learning in such tasks, we develop a method that leverages physical intuitions efficiently solve physical reasoning tasks.
Researchers have developed several benchmarks for evaluat- ing the physical reasoning capabilities of AI agents (Melnik et al., 2023). These benchmarks range from predicting the stability of stack of blocks from an image to recognizing violations of physical principles. on single-intervention physical reasoning tasks, where an agent chooses what action to take at the beginning of an episode and then observes the action's 's effects on the envi- ronment. We focus on the Virtual Tools (Allen et al., 2020) and PHYRE (Bakhtin et al., 2019) benchmarks, which de- spite looking deceivingly simple in their 2D form, are very hard to solve even for humans (Allen et al., 2020). These tasks involve complex physical interactions between objects, which without complete information about the environment dynamics make it impossible for agents to plan the exact solution without active exploration. Agents must interact with the environment to actively obtain information about the object dynamics. The main challenge is to reason about the causal consequences of one's actions, to leverage this knowledge in order to make effective decisions about what actions to try, and to rapidly learn from previous attempts.
this work, we focus
1. Introduction
Consider trying to solve physical reasoning puzzle from the Virtual Tools benchmark, like those shown in Figure 1. Your goal is to have the red ball end up in the green region. To make that happen, you need to choose one the three objects from the top left, place somewhere in the scene, and let gravity and causality do the rest. Which object would you choose and where would you place it? Research in cognitive science suggests that humans solve tasks like these by building internal models of the physical world (Battaglia et al., 2013; Smith et al., 2018; Ota et al., 2021; Zhou et al.,
DDepartment of Computer Science, Stanford University, CA, USA 'Department of Psychology, Stanford University, CA, USA
We propose Causal-PIK, method that leverages physical intuition and causality to solve single-intervene reasoning tasks in as few trials as possible. Our method uses Bayesian optimization to reason about causality via a Physics-Informel Kernel to obtain an expressive posterior distribution over the environment dynamics. This allows an agent to efficiently explore the search space and intelligently
physical
'Department of Computer Science and Technology, University of Cambridge, Cambridge, UK. Correspondence to: Carlota Parés- Morlans <epares @ stanford.edu>.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Causal-PIK: Causality-basee Physical Reasoning with Physics-Informed Kernel
Puzzle
Runs
Puzzle
Runs
solved
solved
(d)
solved
solved
(o)
solved
solved
(c)
(f)
Virtual Tools
PHHYRE
Figure Example puzzles from the Virtual Tools (Allen al., 2020) and PHYYE (Bakhtin et al., 2019) benchmarks. Virtual Tools, the objective is to place one of the blue action objects from the left to have red ball fall into the green area. For PHYRE, the objective place ball of variable radius highlighted in red are failed attempts and the runs highlighted in green successful attempts.
environment have the green and blue objects touch at least 3 seconds. The runs
select what actions to evaluate next. Causal-PIK minimizes the number of physical interactions required. demon- strate that Causal- PPKK significantly outperforms state-of- the-art models on the single-interventin physical reasoning tasks from the Virtual Tools (Allen et al., 2020) and PHYRE (Bakhtin et al., 2019) benchmarks, finding solutions in the fewest number of attempts.
physical intuition into kernel updates during Bayesian opti- mization. We show that our method solves PHYRE fewer attempts than any of these prior methods.
The 'Sample, Simulate, Update' model (SSUP) (Allen et al., 2020) attempts to solve the Virtual Tools benchmark. It samples actions from an object-basee prior, simulates the sampled actions in a noisy physics engine to find the best action to try, executes the action, and updates the model's belief using information from both simulation and execution. SSUP uses a Gaussian mixture model for guidance, but doesn't 't embed information about how actions related via their effects, making SSUP search less efficient. In contrast, our work builds on active learning methods, such as Bayesian optimization, and successfully incorporates additional guidance for action selection by introducing a Physics-Infee Informed Kernel. This significantly lowers how many trials are needed to succeed.
2. Related Work
Physical Reasoning: Researchers have developed several benchmarks towards the goal of improving physical reason- ing capabilities in machine-learning models (Melnik et al., 2023). We evaluate Causal-PIK on the PHYRE (Bakhtin et al., 2019) and Virtual Tools (Allen et al., 2020) bench- marks. Both benchmarks use a physics simulator and cap- ture a variety of complex interaction mechanisms. Further- more, both contain large set of puzzle variations, allowing study generalization across diverse environments.
Bayesian optimization: Causal-PIL uses Bayesian opti- mization (BO), global search method for optimizing black- box functions (Shahriari et al., 2015; Wang et al., 2023). Similar to how we use BO to efficiently find puzzle-solving actions, BO has been used in robotics to reduce the num- ber of trials needed to be run on real robots (Feng al., 2015; Calandra, 2017: Jaquier al., 2020; Berkenkamp et al., 2023). Some prior works proposed to further increase data-efficiency by incorporating simulation-based informa- tion (Antonova et al., 2017; Marco et al., 2017; Antonova et al., 2019). However, they mostly considered continuous parametric controllers, while we need to accommodate a hybrid continuous/diisrett action space.
Both PHYRE and Virtual Tools have inspired several meth- ods for solving physical reasoning puzzles. A number of works (Harter et al., 2020; Girdhar et al., 2020; Ahmed et al., 2021; Qi et al., 2021; Li et al., 2022) that evaluate on PHYRE leverage forward prediction models, also called dynamics models or world models, predict the outcomes of actions. They dynamics models to actions and then execute the most promising actions until one succeeds. Like these works, Causal-PIt also uses dynamics model to reason about the action outcomes. However, unlike these prior methods, Causal-PIK uses observations from previ- ous trials to inform future action selection via Bayesian optimization. Instead of using a dynamies model di- rectly choose actions, use dynamics predictions to instill
Inferring Causality: For the type of tasks we study in this work, the ability to reason about the outcome of an interac-
2

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Causal-PIK: Causality-baxee based Physical Reasoning with a Physics-Informed Kernel
episode terminatedd
The
Learned Dynamics Model
SUCCESS
The
Environment
causel simiartty
randarty
Physics-Informes
at 1/
Causality-Bann Based Action Selection
Figure 2. An overview of Causal-PIK. The Causality- Based Action Selection module proposes the best action to execute based on the predictions from the Gaussian Process. These predictions are generated considering the rewards from previous actions and the causal similarities between candidate actions computed by the Physics-Informed Kernel. These similarities are derived from the individual effects of actions, as predicted by the learned dynamics model. They expressed the change of object O caused by action a, denoted as 0,a
tion critical (Ullman al., 2018; Bramley et al., 2018). Some works aim reason about causality by focusing the Violation of Expectations (VoE) paradigm (Smith et al., 2019; Riochet et al., 2021), where models learn physical plausibility by observing scenes that either follow or violate intuitive physics rules. These models can then judge if new scenes are plausible or not based on past observations. Other works utilize deep neural networks to predict the outcome of interactions from frames (Duan al., 2022; Ye et al., 2018). If an agent can reason about the outcome of inter- actions before actions are taken, then the reasoning can be used to guide smarter action selection, which is beneficial when taking actions in the real world is costly. (Battaglia al., 2013) and (Wu al., 2017) introduce methods that conduct physical reasoning with the help of simulation after constructing representation of the physical world from visual inputs.
state 8o, the future states $1....T are function of the action, can write the score function function of S(D 80. .)) = far)
Given an initial state 80, we aim to solve the task by finding the action that achieves the highest score, namely maximiz- ing (ax), in as few attempts as possible. However, the task's unknown dynamics make it difficult to find optima of fx)). Bayesian optimization find actions that maximize the score.
Therefore,
Bayesian Optimization (BO): The goal of BO is to find the optimal xx that maximizes given objective f(x). The objective captures characteristics of the desired outcome, e.g., our setting, the task score. BO begins with prior that expresses uncertainty over f(x). After evaluating an æ, BO constructs posterior based on the observed data y obtained so far. It then uses auxiliary acquisition function to pick the next to evaluate, taking into account both the posterior mean and covariance. A Gaussian process (GP) is commonly used to model the objective function: f(x) ~ GP(((x) k(-:-).. Its kernel function describes the covariance between the objective values of any pair of input points: k( a:2) cool (xx)). (a:2))
3. Method
Problem Formulation: We consider single-intervention physical reasoning tasks where an agent executes an action the beginning of episode, observes the action's 's effects on the environment over T episode steps, and then receives score at the final timestep. After each episode, the envi- ronment resets, and the agent may attempt the task again by trying another action. Formally, on each attempt, an agent executes an action æ at the first state Su and observes this action's is effect on the environment's state at the remaining timesteps, 8, for 2, The state evolves according to the environment's unknown dynamics, which denote as the function D(80, 2²), to give $1...T = D(so. æ). At the end of the episode, the agent receives score y, which is computed based on the observed states 81.... using a
A common choice for the kernel is the Radial Ba- sis Function (RBF): kRBF(E1, Ø2) = o )² ddi æ2)), where of I can be tuned au- tomatically. The kernel plays crucial role in BO, as encodes an inductive bias about the properties of the under- lying objective function to choose the next æ. The RBF ker- nel, in particular, assumes that values geometrically closer in the input space have higher correlation (are more similar) than values geometrically farther in the input space.
Selecting an appropriate kernel significantly impacts the accuracy of the objective function approximated by the GP,
score function (() = ((s....T). For given initial
3

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Causal-PIK: Causality-basce Physical Reasoning with a Physics-Informed Kernel
thus influencing the performance of BO. We develop a ker- nel that encodes an intuition of causality and physics, which we show solves physical reasoning tasks in fewer attempts than the RBF kernel. In the following section, we describe our full physics-informed BO algorithm Causal-PIK.
helps the GP build noisy prior that includes different areas.
Physics-Informed GP Update (Alg. 1 L4): On the ith attempt, we update the GP surrogate function with all at- tempted actions X and their observed outcomes using our learned Physics-Informed Kernel. Conceptually, the Physics-Informe Informed Kernel encodes similarities and differ- ences between actions based on the predicted effects they will have on the environment. Using this kernel to construct the GP helps the GP use the observed outcomes of already- attempted actions to more accurately predict the outcomes of unexplored actions. We derive the Physics-Informor Kernel in Section 3.2.
3.1. Causal-PIK
Causal-PIK uses BO and reasons about causality with a Physics-Informed Kernel to solve physical reasoning tasks. iteratively proposes actions and uses their observed out- comes, in conjunction with an intuition of physics, to inform future action proposals. We encode physies intuition, such as effects of geometry, mass, and momentum, into the BO framework via the learned Physics-Informed Kernel used to build the GP that approximates the objective function. We will show that incorporating knowledge of physics through the kernel function allows BO to more quickly discover promising regions of the action space.
Causality- Basel Action Selection (Alg. 1 L5): On the ith attempt, we select the next most promising action xf by find- ing the action that maximizes an Upper Confidence Bound (UCB) acquisition function. This acquisition function con- siders the mean and uncertainty of the physics-informed GP. First, we use a Sobol sequence generator to sample a set of naaadddate = 500 candidate actions. Then, we evaluate the acquisition function at each of these 1leandidatc actions. Adopting the intuitive physies procedure proposed by Allen et al., we approximate the outcome of the Thest didate actions with the highest acquisition function values using probabilistic simulation of the task, As per (Allen et al., 2020), approximating the outcomes of actions mim- ics how humans use mental representations to imagine the potential effects of actions before committing to an action. Finally, from this set of Roess actions, we select the action with the highest expected outcome as the next action.
Algorithm 1 Causal-PIK
1: Attempt i 0
2: X = {a: .... Tonng J: y= {91 Venin
can-
3: while success == False do
4: gp + GP GP y, www.pyyyccssoooooeeeeeeeeee.
5: x' +- Causa i tyHasedAct onSelect (gp)
6: y', successs & Execute(a')
7:
X < concat( x'). y concatuy.
8:
i+=
9: end while
Algorithm 1 outlines Causal-PIK. First, we construct initial sets of actions X and scores y to initialize the GP prior us- ing a probabilistic intuitive physics engine (Battaglia et al., 2013). After initializing X and y: we begin the BO proce- dure, iteratively updating the physics-infommed GP surrogate function, choosing the next action and executing it until the puzzle is solved.
We execute the selected action x', observe its outcome, and compute the correspond- ing reward y'. If x solves the task, the algorithm terminates. If a fails to solve the task, we append it along with the score y² to X and y, respectively.
Action Execution (Alg. L6):
3.2. Physics-Informee Kernel
GP Initialization (Alg. 1 L2): To initialize the GP for both initial data
In this section, we describe the Physics-Informed Kernel used to construct the GP in BO (Alg. L4). We construct the Physics-Informe Informed Kernel such that it captures two types of physics intuition that important for solving physical reasoning tasks. The first type is the ability to reason about the causal effect of individual actions, or in other words, the changes that single actions induce in the environment. To capture this, we train a dynamics model to predict future states of the environment given an action and initial state. The second type of physics intuition that we encode into the kernel the ability to reason about causal similarity, namely which actions cause similar changes in the environment. To capture this, we define function that computes how similar two actions are based on their causal effects.
Virtual Tools and PHYRE, we use Ninitsal points. This is the same number of initial data points that our Virtual Tools baseline SSUP (Allen al., 2020) uses to initialize their method. SSUP found that this value provided good trade-off between the number of initial points, total attempts required, and convergence time. Since we use the same initialization framework as SSUP, we expect their parameter analysis to extend to our results. Importantly, like SSUP, we treat these initial noisy rollouts as warm-up samples that do not count towards the total attempt count.
We choose the initial points following the same approach as SSUP: randomly select a dynamic object from the envi- ronment and sample a point from a Gaussian distribution centered at the object's center. As a result, each puzzle attempt has a unique set of Menitial points. This heuristic
As shown in Fig. 2, the Physicc- Informed Kernel function
4

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Causal-PIK: Causality-basee Physical Reasoning with a Physics-Informed Kernel
(a) Cosine Similarity s sim cos (&" )0)
(b) Magnitude Similarity
(c) Causal Similarity
im mag (&", ;',)
sim csl (a, b)
d
g
g
g
0
++0)= = 0
8²
$
g
Figure 3. Illustration of how causal similarity is computed. We obviate the object superscript as there is only one object (red ball) this example. (a): Cosine similarity (Eq. 3). s² has high cosine similarity with s² as they point in the same direction. & points in the opposite direction of s', obtaining negative similarity value. (b): Magnitude similarity (Eq. 4). & has a magnitude that 40% smaller than 8' receiving a low magnitude similarity score. 8² has magnitude that is very close to a', obtaining a high similarity value. (c) Final causal similarity comhining cosine and magnitude similarities (Eq. 6). We obtain medium similarity score for &² relative to a² and low score for Sa relative s²
first uses the learned dynamics model to predict action out- comes. It then uses these dynamics model predictions to predict the similarity between actions. Using the predictions from the learned dynamics model to predict action similarity allows the Physics-Informed Kernel function to model the correlation between actions. In the following sections, we describe the learned dynamics model and the causal sim- ilarity function. Then, we show that the causal similarity function is a valid kernel function.
which captures how similar actions are based on their ability to cause similar outcomes in the environment. To achieve this, we begin by finding the timestep of the first causal event, where the action object interacts with any of the D dynamic objects, which we call fevemt If the action object does not interact with any dynamic object during the Rpred steps, we set levent to be the initial timestep, O. Next, we quantify the causal effect of an action on a dynamic object O based on its predicted motion as
Predicting Causal Effects via Learned Dynamics
(lown +At)
First, we train a dynamics model D to predict the causal effects of individual actions. The physical reasoning tasks we consider involve placing an action object into an envi- ronment with D dynamic objects. The action object and all dynamic objects move the environment according to the unknown world's dynamics. We train the model to predict the effects that individual actions æ have on these dynamic objects. Formally, at each timestep t of an episode, the environment's ' state vector s, consists of the state of the action object, sf², and the states of every dynamic object, s! .,.,.., sp. For given action x and initial environment state 80, the dynamics model D predicts the state at the next Upred timesteps:
At
which computes the state change of object O between Levent and At timesteps after fevent.
To quantify similarity of two actions a and b, we first look at the difference in the state changes, predicted by the dy- namics model, they cause to a given object O. In computing this per-object similarity, we account for both the difference in the directions of the state changes caused by each action on the object (the cosine similarity).
g0.a
OO.b
simm,s (50.a g..bb
11-
(3)
|
and the difference in the magnitudes of the state changes caused by each action on the object,
[8]
8 Spyees D(S0, x).
The Causal Similarity Function
simoag (80,a 8 ,0.b) 1
E |0, 1],
|0.1||
We use predictions from the learned dynamics model [81 Snyaa] to compute the causal similarity of actions,
(4)
5

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Causal-PIK: Causality-based Physical Reasoning with a Physics-Informed Kernel
where g0.a and 80,b are the state changes of object O caused by actions a and b, respectively (as computed by
de min.= di 89) to the goal state achieved at any timestep & in an episode to compute the objective function f(x) (where dist(-) denotes distance function):
Eq. 2). Combining these, the full per-object similarity of actions a and b for an object O is
= { dist(su, 8o) ( ((1 de
[(x) =
exp(Sde) if de < dist(su,
simmay (O, a, b) max [
0, simeos (50m,, 8..b) slmmmng (00,,o0 E [0.
0
otherwise
(5)
3.4. Leveraging Causality for Efficient Sample Learning
Finally, we define the full causal similarity metric between actions and b as:
Unlike methods that learn only from direct observations, our approach leverages physical reasoning to infer the outcomes of untested actions that are predicted to share the same causal effect as observed ones. As a result, a single rollout allows our model to update its belief not just about the executed action, but also about actions that are predicted to produce a similar physical outcome. This significantly enhances sample efficiency and enables reasoning about alternative scenarios without exhaustive exploration.
D
simeat (a, b)
1
Σ
si (O, a, b)]
D
0=1
D
simoryy (O, (,. 00 )
Σ O=1
exp
(6)
The above computes the mean per-object similarity over all D dynamic objects scaled by the exponentiated mean to accentuate differences between state changes caused by actions. Figure 3 provides a visual example of Equations (3)-(6) and shows how the causal similarity metric captures both directional and magnitude similarity.
Our Physics-Informed Kernel explicitly encodes causal de- pendencies between actions and their physical consequences. In contrast to standard approaches that tend to cluster actions based on proximity in the feature space, Physics-Informet Kernel compares them by assessing their impact on the envi- ronment. Using counterfactual reasoning, the kernel distin- guishes between causal effects those directly attributable to actions and confounding factors arising from the dy- namics of the environment.
From Causal Similarity to Physics-Informed Kernel
The causal similarity function in Equation 6 uses predicted state changes to quantify the correlation between actions that have similar causal effects. Consequently, these scores possess significant potential to be used the kernel fune- tion of GP that models the objective function of physical reasoning tasks. For a function to be a valid kernel, it must satisfy two fundamental properties: symmetry and positive semi- semi-definiteness. Analyzing Equations (3)-(6), the reader can verify that sa, b) = sim (b, a), thereby fulfill- ing the symmetry criterion. Furthermore, for any arbitrary pair of actions and b, their similarity score sim (a, b) is always non-negative. As such, causal similarity satis- fies both criteria of valid kernel function, so we define the Physics-Informed Kernel used in Algorithm Line 4 as simay(a, b).
This distinction is achieved by evaluating action outcomes against a counterfactual baseline where no action object is placed in the environment. In Equation 7, the shortest observed distance de normalized by the counterfactual baseline distance dist( so, 89). Similarly, when computing the causal similarity of actions in Equation 6, the counter- factual baseline helps cluster actions that have no effect on the environment and emphasizes the attributable effects of other actions.
4. Experimental Setup
Benchmarks:
3.3. Causal-PIK objective function
We test Causal-PIK on the 20 original puz- zles from the Virtual Tools benchmark. These puzzles in- volve placing an action object in a 2D dynamic physical environment to guide a red ball into the green goal area. Once an action object is placed, gravity is activated, and its effect can be observed.. The three-dimensional action space consists of: Ipos E [0. 600). Mpoos E [0, 600). and actionatj E {tool,, lool2, tools). resulting in a total of 1,080,000 possible actions.
Virtual Tools:
We use Causal-PIK to find an optimal action æ that max- imizes the objective function (x). On each attempt, we use an objective function (() that quantifies the progress that executing action æ makes towards a goal state sº. We find that for tasks with complex dynamics, is important for the objective function to capture how close action gets to reaching the goal state at any timestep in the T- timestep long episode, not just at the final timestep. This insight aligns with the concept of "almost" reaching a goal, introduced by Gerstenberg et al. (Gerstenberg & Tenen- baum, 2016). To capture this, we use the closest distance
PHYRE: We test Causal-PIK on the 25 puzzles from the BALL tier in PHYRE, following the gross-generalizaatin
6

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Causal-PIK: Causality-based Physical Reasoning with a Physics-Informed Kernel
Fraees Pec
f())
success
x
SUCCEsS
Fune (an
JaB (z)
Figure 4. Comparison of posterior distributions (az) generated by Causal-PIK and a BO agent with an RBF kernel. Given the same initial observations {a' ) Causal-PIK produces more expressive posterior, effectively clustering actions based on their causal effects the red area contains those actions predicted to move the ball closer the goal while the blue contains those predicted push it further away. In contrast, the BO agent with the RBF kernel requires more iterations to develop an informative posterior.
setting. These puzzles require selecting the radius of ball and placing it in 2D dynamic physical environment to have the blue and green objects stay in contact for at least seconds. As in Virtual Tools, gravity is activated upon placement, after which the action's i effect observed. The three dimensional action space consists of: Tpos E [0, 256). Upos E [0, 256], and actionabj [2, yielding total of 2,555, 5,44 possible actions.
Allen et al. (2020). Here, we conducted human experiment with PHYRE by recruiting n = 50 participants from Prolific and asking each participant to solve one variation of each of the 25 puzzles. Participants had 10 attempts to solve each puzzle by using the mouse to draw and place a valid action object. As with the model setup, each puzzle was initially presented freeze-frame, and then gravity was activated once the participant submitted their action (see Appendix B for details). The experiment was approved by the Stanford Institutional Review Board (IRB).
Baselines and Ablations: For Virtual Tools, we compare Causal-PIK against SSUP (Allen et al., 2020), the state-of- the-art method for this benchmark. We also compare against the two naive baselines from Allen et al., RAND and DQN. For PHYRE, we compare Causal-PIK against three cate- gories of prior work. The first category consists of a method that uses reduced action space of size 1,000 (Girdhar et al., 2020). The second category consists of methods that use reduced action space of size 10,000 (Ahmed et al., 2021; Qi ct al., 2021). Critically, by reducing the action space from ~25 million actions, these methods simplify the problem dramatically. The third category consists of methods that operate over the full action space (Harter et al., 2020), we do. Additionally, lines from the original PHYRE paper (Bakhtin et al., 2019): RAND, MEM, and DQN. Finally, to assess the impact of the Physics-Infee Informed Kernel, we conduct an ablation study by replacing with a standard RBF kernel. This baseline highlights the expressiveness of the Physics-Inforee Ker- nel, particularly in tasks with complex dynamics, where capturing underlying physical principles is key to finding solutions efficiently.
We measure the performance of our method using the AUCCESS metric introduced by Bakhtin et al.. This metric aggregates the success percentages via weighted average, placing more emphasis solving tasks with fewer attempts. AUCCESS is computed as follows:
Evaluation Metric:
Ex Wk where
AUCCESS
Σx wk
Ws = log(k: + 1) - log(k), for E f1,.... MAX Sk percentage of tasks solved within ie attempts
(8)
report results for three naive base-
We compare overall AUCCESS scores for each model, as well as correlation between scores for humans and models across individual puzzles.
Dynamics Model: We use Region Proposal Interaction Networks (Qi al.. 2021) as the architecture. We train the model on puzzles that share underlying physical con- cepts similar to the test puzzles. For Virtual Tools, we train the model on 10 variations for each original puzzle. For PHYRE, for each of the 10-fold splits from Bakhtin et al., we train a model exclusively on the fold's training set, ensur- ing that Causal-PIK' is tested on previously unseen puzzles. At inference time, image the test puzzle with the action
Human Baseline: addition to model baselines, we com- pare Causal-PIK against a human baseline for both bench- marks. A human study with Virtual Tools was conducted by
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Causal-PIK: Causality-basct Physical Reasoning with a Physics-Informed Kernel
embedded and the initial bounding boxes of all dynamic objects in the scene fed into the model. The model out- puts the bounding boxes of the next Rpreed time steps. We set Mpredd to capture the initial steps, but not the full roll- out. The choice of Rppred helps the dynamic model focus on learning the immediate causal effect of an action, which is then used to compare the similarity between actions. For further details, refer to Appendix A.
Table 1. AUCCESS scores of Causal-PIK, its RBF ablation variant, and state-of-the orrthe-rrt model on the Virtual Tools benchmark. Results are based on a maximum of 10 attempts per task, with higher scores indicating more efficient problem-solving.
Model
AUCCESS
RAND
16.0+20.0
DQN
25.0+24.0
SSUP (Allen et al., 2020)
58.0
5. Results and Discussion
Ours RBF
42.0
Virtual Tools: We evaluate Causal-PIK on the 20 Virtual Tools puzzles with 100 tests per puzzle. For each test, we limit the maximum number of attempts to 10. Table in- cludes summary of the AUCCESS rate achieved by the different agents. Causal-PIK is 7 points higher than the best- performing baseline, requiring lower number of attempts to solve the tasks.
Ours Causal-PIK
65.0
Humans (Allen et al., 2020)
53.25
Table 2. AUCCESS scores of Causal- its RBF ablation variant, and state-of-the-art models on PHYRE-IB Cross. Results are based maximum of 100 attempts per task and averaged over 10 test folds. Higher scores indicate more efficient problem-solving.
We evaluate Causal-PIK on the PHYRE-ID 1B Cross test set, which comprises 25 tasks distributed across 10 folds, with 10 variations per task and 10 tests per fold. For each test, we limit the maximum number of attempts to 100. As shown in Table 2, Causal-PIK achieves an AUCCESS rate over 10 points higher than the best- best-performing baseline. We also compare our results to baselines that use reduced action space (Girdhar et al., 2020; Ahmed et al., 2021; Qi et al., 2021), which is guaranteed to lead to a 100 AUCCESS score for 10K actions if an optimal oracle is used.
PHYRE:
Model
AUCCESS 1
Dec [Joint] (Girdhar et al., 2020)*
40.3
MEM
18.5+5.1
DQN'
36.8
Ahmed et al. 2021
41.9+8.8
RPIN (Qi et al., 2021)!
42.2+7.1
RAND
13.0+5.0
Harter et al. 2020
30.24
As shown in Table Causal-PII performs comparably to baselines that utilize drastically reduced action space. Our approach tackles a significantly harder problem by consid- ering any point in the action space. Bakhtin et al. (2019) analyzed how the number of actions ranked by agents at test time affects performance. As shown in Figure 4 of their work, the AUCCESS of the DQN agent decreases as the number of ranked actions increases by orders of magni- tude, up to a maximum of 100,000, which still far from the 2,555,900 5,04 possible actions per puzzle in the full action space. We argue that discretizing the environment for action selection is an unrealistic constraint when aiming to develop generalist algorithms capable of solving complex physical reasoning tasks.
Ours RBF Ours Causal-PIk
27. 27.70
41.6--..33
Ours Causal-PIL @10+ Humans @ 10+
24.8_9.22
36.6_10.2
"IK reduced action space. 110K redueed action space. max of 10 attempts per task.
red area on the top left side of the red ball, clustering actions that move the ball closer to the goal: a blue area on the top right, clustering actions that push it further away; and a yellow area representing unexplored actions that cause no movement. In contrast, the BO agent with the RBF kernel requires more steps to build an informative posterior.
Physics-Informen Kernel versus RBF Kernel: We com- pare the posterior distributions of Bayesian Optimization (BO) agent using the Physics-Intie Informed Kernel and the RBF kernel. As shown in Figure 4, given the same set of initial ob- servations, the BO Agent with the Pnysree Kernel produces a more expressive posterior that focuses on high- likelihood actions. Given two actions, one that moved the red ball further from the goal and another one that brought it closer but overshoots, the GP with the Physics-Informer Kernel constructs a posterior with three distinct regions:
Participants found the puzzles from both benchmarks to be challenging. Causal-PIK achieved a higher AUCCESS score than humans on both benchmarks, except when it was restricted 10 attempts in PHYRE. The high variance in AUCCESS rates for both humans and models suggests that the puzzles had mixed levels of dif- ficulty. We computed AUCCESS scores across individual puzzles. On Virtual Tools, human per-puzzle scores were most correlated with SSUP (r = 0.71), followed by Causal- PIK (r = 0.63), then DQN (r = 0.32). Causal-PIK may be less correlated with humans but still have a higher overall
Human performance:
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Causal-PIK: Causality-basce Physical Reasoning with a Physics-Informed Kernel
AUCCESS rate because it is able solve several puzzles that humans find very difficult. This lowers the per-puzzle correlation, but highlights the overall performance of our method. On PHYRE, human scores were most correlated with Causal-PIK (r = 0.73), followed by Causal-PIK @ 10, which has limited attempts (r = 0.71), then the RBF ker- nel baseline (r = 0.64), and finally Harter et al. (2020) (r= 0.55). Overall, the high correlation in scores between humans and our model, even when restricted to maximum of 10 attempts per puzzle, suggests high alignment in the types of physical dynamics that were found to be easy or difficult to reason about.
6. Limitations
Causal-PIK currently does not share knowledge across tasks. Enabling agents to recognize similarities between tasks and leverage past observations from tasks requiring similar phys- ical reasoning remains an avenue for future work. By iden- tifying regions of the action space that share underlying dynamics, agents could integrate prior knowledge to solve new tasks more efficiently.
Another limitation arises from the noise introduced by causal effect predictions, which directly impacts perfor- mance. Poor predictions introduce misleading similarities, potentially guiding the agent in the wrong direction. Im- proving these predictions would improve the expressivity of the Physics-Ifformed Kernel. However, our results demon- strate that Causal-PIK remains robust despite this noise, suggesting potential for future sim-to-real transfer.
Resistance to noisy predictions We conduct an analysis to study the impact of the accuracy of the dynamics model on the performance of our method. To demonstrate that existing predictions are inherently noisy due to the characteristies of the train and test sets, we train the PHYRE dynamics model on tasks from the test templates, ensuring prior exposure to similar puzzles. As result, the L2 error for object bounding boxes improved to 3,56, compared to 19.3 = 4.55 when tested on entirely unseen puzzles.
Additionally, the physical reasoning tasks considered in this study involve three-dimensional action space. Scaling to nigher-dimensione action spaces would require modifi- cations to the causal effect predictor to accommodate the added complexity. However, the kernel equations (2-6) would remain unchanged, as the new dimensions would be encoded within the state returned by the model. Thereby, the fundamental structure of our approach remains unchanged, preserving the kernel's ability to compare the immediate effects of high-dimensional actions. Moreover, Bayesian Optimization (BO) is expected to remain effective in larger search spaces, as prior work has demonstrated its robustness in high-dimensional action spaces in real-world applications (Antonova et al., 2019).
Despite this difference in prediction accuracy, our method achieved an AUCCESS of 45, which is only 4 points higher than the 41.6 = 9.33 AUCCESS we reported for the case with unseen dynamics. This demonstrates that even with a substantial increase in prediction error, the performance drop is small, indicating that our method remains resilient to noisy dynamic predictions. While improved dynamic predictions can enhance performance, our approach does not rely on perfect predictions, retaining robustness even in the presence of inaccuracies.
On the shortcomings of RL agents: Reinforcement learn- ing (RL) agents like DQN have shown remarkable perfor- mance in strategic games such as Atari and Go after exten- sive training, but they struggle in tasks that require physical reasoning. In the context of the Virtual Tools and PHYRE physical reasoning tasks, DQN fails to generalize effec- tively to unseen puzzles due to its limited understanding of the physical properties of objects and the basic rules gov- erning their interactions. Previous studies on the PHYRE and Virtual Tools benchmarks (Li et al., 2024; Allen et al., 2020) have underscored this limitation, demonstrating that RL agents struggle to understand the underlying physics involved beyond their strong mapping between states and actions. This shortcoming leads to broad and inefficient unguided exploration process, requiring numerous attempts to discover promising areas. Furthermore, DQN fails to learn from past attempts, often repeating similar unsuccess- ful actions. In contrast, our approach focuses on learning from each failed attempt and uses the causality and phys- ical insights from the Physies-Informed Kernel to guide exploration towards more promising areas.
7. Conclusion
We introduce Causal-PIK, novel approach that integrates a Physics-Informed Kernel with BO to reason about causality in single-intervention physical reasoning tasks. By lever- aging information from past failed attempts, our method enables agents to efficiently search for optimal actions, re- ducing the number of trials needed to solve tasks with com- plex dynamics. Our experimental results demonstrate that Causal-PIK outperforms state-of- the-ahr art baselines, requir- ing fewer attempts on average to solve the puzzles from the Virtual Tools and PHYRE benchmarks.
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Causal-PIK: Causality-basee Physical Reasoning with a Physics-Informed Kernel
Acknowledgments
Calandra, R. Bayesian Modeling for Optimization and Con- trol in Robotics. PhD thesis, Darmstadt University of Technology, Germany, 2017.
We thank Kelsey Allen and Kevin Smith for their insight- ful discussions and valuable feedback. We also thank the reviewers for their thoughtful feedback, which helped im- prove the quality of the manuscript. The Stanford Insti- tute for Human-Centered Artificial Intelligence (HAT), the Sloan Research Foundation, and Intrinsic provided funds to support this work. TG was supported by a grant from Cooperative AL.
Duan, J., Yu, S., Poria, S., Wen, B., and Tan, C. Pip: Physi- cal interaction prediction via mental simulation with span selection. In European Conference Computer Vision, pp. 405-421. Springer, 2022.
Feng, S., Whitman, E., Xinjilefu, X., and Atkeson, C. G. Optimization-basee Full Body Control for the DARPA Robotics Challenge. Journal of Field Robotics, 32(2): 293-312, 2015.
Impact Statement
This paper presents work whose goal is advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Gerstenberg, T. and Tenenbaum, J. Understanding" almost": Empirical and computational studies of near misses. In CogSci, 2016.
Girdhar, R., Gustafson, L., Adcock, A., and van der Maaten, L. Forward prediction for physical reasoning. arXiv preprint arXiv:2006. 10734, 2020.
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2023.

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Causal-PIK: Causality-basee Physical Reasoning with a Physics-Informed Kernel
A. Dynamics Model
train the dynamics models to predict causal effects of actions in the Virtual Tools and PHYRE benchmarks (Allen et al., 2020; Bakhtin al., 2019). Our model is based on the Region Proposal Interaction Networks (RPIN) architecture (Qi et al., 2021) and is trained on puzzles that share similar underlying physical principles with the test puzzles.
For the Virtual Tools benchmark, we generate 10 variations of each of the 20 original puzzles, modifying object sizes and relative positions to create diverse scenarios (see Figure 5). None of the original puzzles are included in training. For each puzzle variation, 300 actions are generated, distributed evenly across the three action object types. At least 50% of actions result in collisions between the action object and a dynamic object, while 10% simulate the absence of an action object in the environment. The remaining actions are randomly sampled using a Sobol generator to determine object placement. The inclusion of "no-action" cases enables the model to implicitly learn that stationary objects remain static when no net external force is applied.
For the PHYRE benchmark, train 10 separate dynamics models, per fold. Each model is trained on 20 out of the 25 puzzles assigned to the training set for that fold. For each puzzle, we generate 500 actions, of which 350 result in failed rollouts, 150 result in successful rollouts, and 50 simulate the absence of an action object. Actions are randomly drawn from the 100,000 pre-selected actions provided by (Bakhtin et al., 2019).
At inference time, the model receives an image of the puzzle, along with the initial bounding boxes of all dynamic objects and the action object. The model outputs the bounding boxes for the next 7ppred time steps. We rppred to 20, which usually captures one collision but not the full roll-out. The choice of Prredd helps the dynamic model focus on learning the immediate causal effect of an action, which is then used to compare similarity between actions.
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Figure 5. Puzzle variations for one the original puzzles the Virtual Tools benchmark. The puzzle variations share similar underlying physical concept with the original puzzle, but have different dynamics due to varying object sizes and relative positions.
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Causal-PIK: Causality-basee Physical Reasoning with a Physics-Informed Kernel
B. PHYRE Human Experiment
We adapted the PHYRE-IB benchmark into a suite of online games using Planck.js (Shakiba, 2017), JavaScript rewrite of the Box2D physics engine used in PHYRE (Bakhtin al., 2019). The experiment was posted on Prolific, an online crowd-sourcing research platform. We recruited
50 participants (age: M 37, SD 11; gender: 20 female, 27 male, I non-binary, and undisclosed; race: 29 White, Black, 5 Asian, 1 American Indian/Alaske Native, Multiracial, and 2 undisclosed) and compensated them at rate of $12/hour. All participants were native English speakers residing in the US.
Participants were first given instructions for the task, including a "playground" environment where they learned about the different objects and the dynamics the world. They then solved simple practice puzzle before moving on to the main puzzles. During the main portion of the experiment, they were given one variation each the 25 puzzles in Phyre- in a random order. On each puzzle, participants were initially presented with a freeze-frame of the scene. They attempted to solve the puzzle by using the mouse draw and place the action object red ball) in any valid location in the scene. On each attempt, they watched the simulation run until either succeeded, after which they continued to the next puzzle, timed out or all objects stopped moving, at which point they could try again. If they ran out of attempts (maximum 10), then they were also directed to the next puzzle. We enforced all the same physics parameters and time limit described in the original PHYRE benchmark. We recorded participants actions (Tpos. Upos. and actionabj = Thatt) on each attempt. Overall, participants spent an average of 1.7 minutes (SD = 1.4) on each puzzle.
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Reviews:
[{"review_id": "yqiVM8bVAL", "reviewer": "Reviewer_jF9J", "summary": "The paper introduces Causal-PIK, a novel approach that integrates a Physics-Informed Kernel with Bayesian Optimization to reason about causality in single-intervention physical reasoning tasks. Experimental results on Virtual Tools and PHYRE physical reasoning benchmarks verify the proposed method could finish the task with fewer actions.\n\n## update after rebuttal\nThank you for the thoughtful rebuttal. I appreciate the effort to address my concerns. I will maintain my original score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer jF9J for their thoughtful feedback. We are glad they consider our method to be clearly presented with a sound experimental design which includes good ablation studies. We address the reviewer’s comments and will incorporate all of the following discussions in the final draft.\n\n> [jF9J.1] Scaling to bigger states and actions\n\nWe appreciate the reviewers’ interest in how our approach scales to higher-dimensional action and state spaces. We would like to emphasize that our method already operates in significantly larger action spaces compared to many baseline approaches. For example, in the Phyre benchmark, our action space comprises 2M possible actions, whereas other baselines consider only 10K.\n\nIncreasing the state and action space—such as introducing multiple objects in specific configurations or simultaneously determining both the direction and speed for accurate object throws—does not alter the fundamental kernel equations (2)–(6), meaning the kernel would remain expressive at comparing the immediate effect of these high dimensional actions. Certainly, the search space for BO would increase as these new dimensions are encoded within the state returned by the dynamics model. However, researchers have shown that BO scales robustly with high-dimensional actions (Antonova et al., 2019). We included this information in the limitations section of our manuscript.\n\n> [jF9J.2] Updated PHYRE results for our method (Table 2)\n\nWe appreciate the reviewer's feedback which prompted additional experiments. During this process, we discovered and fixed a bug in our AUCCESS score computation for the Phyre benchmark. This correction has resulted in the following updated PHYRE results in Table 2:\n\n* Ours Causal-PIK: 41.6 ±9.33 (previously reported as 51.3±8.46)\n\n* Ours RBF: 27.7±9.68 (previously reported as 31.99±9.46)\n\nThe Virtual Tools results in Table 1 remain the same, as they were not affected by this bug.\nImportantly, these corrections do not affect the claims or contributions of our paper. Our approach still achieves state-of-the-art performance, maintaining a significant 10-point margin above the baseline that uses the same size action space. Furthermore, our method performs comparably to baselines that utilize a drastically reduced action space of 10K.\n\nWe will incorporate these corrected values in the final manuscript draft.\n\n> [jF9J.3] Tasks lacking complexity\n\nWe appreciate the reviewer’s interest in understanding how our method scales to more complex tasks. While the tasks in this study may appear deceptively simple in their 2D form, they are in fact quite challenging—even for humans. Solving these puzzles requires an understanding of the underlying physics involved, such as momentum, balance, geometry, mass, and propulsion. Importantly, agents do not have any details about the environment, such as object density, friction coefficients or material composition, making it impossible to plan the exact solution without active exploration.\n\nTo further emphasize the complexity of our current setup, we are conducting an ongoing human study using the Phyre benchmark, which is similar to the study presented in Allen et al. (2020). A total of 50 participants (currently n=17) will be recruited from Prolific and shown one variation of each of the 25 puzzles in a random order. Participants will have 10 attempts to solve each puzzle by using the mouse to draw and place the ball in any valid location in the scene. On each attempt, they will watch the simulation run until it either succeeds, after which they will continue on to the next puzzle, or time out or all objects stop moving, at which point they can try again. If they run out of attempts, then they will also be directed to the next puzzle. Preliminary results show that participants spend about 1.8 minutes on each puzzle. Preliminary AUCCESS scores, which will be added to the manuscript to complement Tables 1 and 2, are as follows:\n\n* Virtual Tools - Humans: 53.25 ± 23 (Causal-PIK @10: 65.0 ± 25.0)\n\n* Phyre - Humans: 34.9 ± 10.72 (Causal-PIK @10: 24.8 ± 9.22)\n\nThese preliminary results demonstrate that humans find these puzzles to be very challenging.\n\nA logical progression of our work would involve creating a 3D high-fidelity environment for these tasks. This shift to a more complex setting would still leave the kernel equations unchanged. We would only need to change the dynamics model to one designed for 3D environments, such as those proposed by Xue et al. (2024) [A] or Driess et al. (2023) [B].\n\n[A] Xue, H., Torralba, A., Tenenbaum, J., Yamins, D., Li, Y., & Tung, H. Y. (2024). 3D-IntPhys: towards more generalized 3D-grounded visual intuitive physics under challenging scenes. Advances in Neural Information Processing Systems, 36\n\n[B] Driess, D., Huang, Z., Li, Y., Tedrake, R., & Toussaint, M. (2023, March). Learning multi-object dynamics with compositional neural radiance fields. In Conference on robot learning (pp. 1755-1768). PMLR", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes, though the targeted task of physical reasoning seems to be only a toy setting with only 3-30 object candidates and 2D coordinates.", "theoretical_claims": "There are no theoretical claims.", "experimental_designs_or_analyses": "The experimental designs are sound, with good ablation studies on the proposed PIK kernel. The experimental analyses seem valid.", "relation_to_broader_scientific_literature": "The paper designed a physics-informed kernel and uses Bayesian Optimization to reason over the causality. If the approach should be able to scale to real-world-level tasks requiring more complicated state and action space, it would be helpful.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strength: The paper is clearly presented, and the proposed method achieves new sota on two benchmarks.\n\nWeakness: The targeted task of physical reasoning seems to be only a toy setting with limited action space and oracle state space, whether the proposed method could be useful with real questions remains unknown.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "Is there any evidence the proposed method could scale up to solving physical reasoning problems of more complexity?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8JsAo2voGC", "reviewer": "Reviewer_nzda", "summary": "This paper proposes a method, Causal PIK, using Bayesian optimization for causal reasoning via a Physics-Informed Kernel, in order to obtain an expressive posterior distribution over the environment dynamics. \nUnlike prior works directly using a learned dynamics model to choose actions, Causal-PIK uses dynamics predictions to instill physical intuition into kernel updates during Bayesian optimization.\nA crucial component of the physics informed kernel is causal similarity of actions, capturing how similar actions are based on their ability to cause similar outcomes in the environment.\nThe proposed method is tested on single-intervention physical reasoning tasks: Virtual Tool and Phyre, and is shown to beat state-of-the-art methods on these benchmarks.\n\n\n\n## Update after rebuttal\n\nThanks for the rebuttal. I remain positive about the paper and will maintain my score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer nzda for their thoughtful feedback. We are glad they consider the chosen benchmarks to be relevant for the task at hand. We address the reviewer’s comments and will incorporate all of the following discussions in the final draft.\n\n> [nzda.1] High variance in the results presented in Tables 1 and 2\n\nThe high variance observed in the results presented in Tables 1 and 2, particularly for the Virtual Tools benchmark, can be attributed to the nature of the puzzles themselves. Some puzzles are significantly more challenging, leading to near-zero scores, while others are more easily solvable, creating a broad distribution of results. This effect is not unique to our method—other baselines also exhibit substantial variance for the same reason.\n\nTo further illustrate this, in the Phyre benchmark, the supplemental material (Figure 10 in ((Bakhtin et al., 2019)) provides histograms showing the distribution of the number of actions that solve a given task. These histograms highlight how some tasks are much more difficult than others, reinforcing the idea that the variance is an inherent property of the task design rather than a shortcoming of any particular method.\n\n> [nzda.2] Characteristics and dependence on initial point set\n\nPlease refer to comment [vdey.1].\n\n> [nzda.3] Physics-Informed Kernel limitations\n\nWe appreciate the reviewers' interest in understanding the limitations of the Physics-Informed Kernel (Causal-PIK). As outlined in the limitations section of the paper, Causal-PIK currently does not share knowledge across tasks. Enabling agents to recognize similarities between tasks and leverage past observations from tasks requiring similar physical reasoning remains an important avenue for future work. By identifying regions of the action space that share underlying dynamics, agents could potentially integrate prior knowledge to solve new tasks more efficiently.\n\nAnother limitation arises from the noise introduced by causal effect predictions, which directly impacts performance. Poor predictions can introduce misleading similarities, potentially guiding the agent in the wrong direction. Improving these predictions would enhance the expressivity of the Physics-Informed Kernel. However, our results demonstrate that Causal-PIK remains robust despite this noise, suggesting potential for future sim-to-real transfer. For additional information on how Causal-PIK resists noise in causal effect predictions, we kindly refer the reviewer to comment [d7h6.1].\n\n> [nzda.4] Scaling to bigger states and actions\n\nPlease refer to comment [jF9J.1].\n\n> [nzda.5] Improving Figure 2\n\nWe appreciate the reviewer’s feedback on the action similarity component of Figure 2. We will make this part of the figure clearer in the final manuscript draft.\n\n> [nzda.6] Updated PHYRE results for our method (Table 2)\n\nPlease refer to comment [jF9J.2].", "claims_and_evidence": "The authors highlight the importance of the kernel used for the Gaussian Process and instead of a standard RBF kernel, they develop a kernel that encodes an intuition of causality and physics. They show that this outperforms the RBF kernel.", "methods_and_evaluation_criteria": "The proposed method is evaluated on two benchmarks: Virtual Tool and Phyre.  \nThe main baseline in Virtual Tools is SoTA method SSUP, which samples actions from an object-based prior and simulates the sampled actions to find the best action to try. \nIn Phyre, Causal-PIK operates over the full action space, but also compares against some methods that operate on a reduced action space, which simplify the problem.\n\nThe authors show that their method outperforms baselines on these benchmarks. However, the standard deviation in the results are quite high. That's why I am wondering if the results are statistically significant.", "theoretical_claims": "n/a", "experimental_designs_or_analyses": "The tested benchmarks are valid. My main concern at the moment is the high variance across the results.", "relation_to_broader_scientific_literature": "Physical reasoning models, intuitive physics are well-established concepts. Causal-PIK, unlike prior work, uses Bayesian optimization using previous trials to inform future action selection.", "essential_references_not_discussed": "not to my knowledge", "other_strengths_and_weaknesses": "Strengths:\n- method that can efficiently solve causal intervention tasks with a few attempts via physics-informed Bayesian Optimization\n\n\nSome weaknesses:\n1. The authors first construct initial sets of actions X and scores y to initialize the GP prior using a probabilistic intuitive physics engine. The dependence on this initial dataset is not ideal. \n2. Do the authors see any limitations with the current form of the physics-informed kernel? \n3. How do the authors expect this method to scale in high-dimensional action/state spaces? \n4. How do the authors interpret the high variance in the results presented in Tables 1 and 2? Especially for the Virtual Tools benchmark", "other_comments_or_suggestions": "1. I think Figure 2 can be further improved for clarity, especially the action similarity component.", "questions_for_authors": "Copying over my questions from the strengths and weaknesses field above:\n\n1. The authors first construct initial sets of actions X and scores y to initialize the GP prior using a probabilistic intuitive physics engine. The dependence on this initial dataset is not ideal. Can the authors comment on this part? How many samples are needed? \n2. Do the authors see any limitations with the current form of the physics-informed kernel? \n3. How do the authors expect this method to scale in high-dimensional action/state spaces? \n4. How do the authors interpret the high variance in the results presented in Tables 1 and 2? Especially for the Virtual Tools benchmark", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7tYzhL3ZN3", "reviewer": "Reviewer_vdey", "summary": "The paper attempts to address the challenge of single-intervention physical reasoning tasks. It proposes Causal-PIK, which combines Bayesian optimization and a Physics-Informed Kernel. The method leverages physical intuition and causality to iteratively find optimal actions. Experimental results on the Virtual Tools and PHYRE benchmarks demonstrate that Causal-PIK outperforms previous state-of-the-art approaches. It achieves higher AUCCESS scores, requires fewer attempts to solve complex physical reasoning puzzles.\n\n## update after rebuttal\n\nI have read the authors response. The authors' reply has addressed some of my concerns. And I agree in general with other reviewers that it is a good work and I lean towards acceptance.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer vdey for their thoughtful feedback. We are glad they found the method implementation to be well-executed with a design that effectively captures important physical intuitions. We address the reviewer’s comments and will incorporate all of the following discussions in the final draft.\n\n> [vdey.1] Characteristics and dependence on initial point set\n\nThank you for pointing out that we did not state the number of initial data points and how we chose these data points. To initialize the GP for both Virtual Tools and PHYRE, we use $n_{initial} = 9$ initial data points. This is the same number of initial data points that our Virtual Tools baseline SSUP (Allen et al., 2020) uses to initialize their method. SSUP found that this value provided a good trade-off between the number of initial points, total attempts required, and convergence time. Since we use the same initialization framework as SSUP, we expect their [parameter analysis](https://www.pnas.org/doi/suppl/10.1073/pnas.1912341117/suppl_file/pnas.1912341117.sapp.pdf) to extend to our results. Importantly, like SSUP, we treat these initial noisy rollouts as warm-up samples that do not count towards the total attempt count.\n\nWe choose the $n_{initial}$ initial points following the same approach as SSUP: randomly select a dynamic object from the environment and sample a point from a Gaussian distribution centered at the object's center. As a result, each puzzle attempt has a unique set of random $n_{initial}$ points. This heuristic helps the GP build a noisy prior that includes different areas. \n\nWe will incorporate these relevant details in the final draft.\n\n> [vdey.2] Causal-PIK tested on I-PHYRE\n\nThank you for the suggestion to consider multi-step reasoning tasks like I-PHYRE. While we acknowledge the value of such tasks, we would like to highlight that I-PHYRE is not necessarily a more challenging problem than our current setting. For example, assuming a time resolution of 0.01s and five removable objects over a 15-second period, the action space in I-PHYRE would be limited to 7,500 possible actions—significantly more constrained than the over 2 million possible actions in our setting for PHYRE.\n\nTo the best of our knowledge, no prior work has attempted both benchmarks. While benchmarks like Phyre and Virtual Tools focus on spatial physical reasoning, I-PHYRE emphasizes time-based physical reasoning. Nonetheless, our method could be adapted to solve such problems by querying Causal-PIK at every time step and incorporating a no-op action choice. While this modification falls outside the scope of our current contribution, we believe it could offer valuable insights for future developments.\n\n\n> [vdey.3] Scaling to bigger states and actions and generalizing to more complex scenarios\n\nPlease refer to comments [jF9J.1] and [jF9J.3].\n\n> [vdey.4] Updated PHYRE results for our method (Table 2)\n\nPlease refer to comment [jF9J.2].", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "There is no theoretical claims made in this work. So does not apply.", "experimental_designs_or_analyses": "Yes. I have checked the experimental designs. I do find many choices of the setup are valid and support their claims.", "relation_to_broader_scientific_literature": "The paper extends the existing work of SSUP. But unlike SSUP, it propose a Bayesian Optimization method combined with a physics-informed kernel to learn and suggest new actions on the fly. The method can adapt to the accumulated experience and the results show that this method does help in learning.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strengths\n* The paper fills a gap in the research area as few works have focused on effectively building a model based on past experience for physics tasks. By introducing Causal-PIK, it extends our understanding of computational models for such tasks, offering new insights into how agents can learn from previous attempts to solve complex physical reasoning problems.\n* Implementation of the proposed method is well-executed. The design of the Physics-Informed Kernel is reasonable, as it effectively captures important physical intuitions like causal effects and causal similarity between actions. This kernel is crucial for the performance of the overall method, enabling more accurate predictions of action outcomes.\n* The paper combines advances in neural networks, such as using the RPIN for the dynamics model, and machine learning methods like Bayesian optimization. This integration allows Causal-PIK to leverage the benefits of both fields, resulting in a more powerful approach that can efficiently search for optimal actions in complex physical environments.\n\nWeaknesses\n* The paper does not explicitly state the number of initial data points used in Algorithm 1 for the warm-up phase. It's unclear how many initial data points are needed to reach good performance. Also, it seems to me that the selection of seed data is likely to be critical. Since the method relies on these initial observations to update the model and select subsequent actions, poorly chosen seed data might lead the algorithm way off. Would it be possible to show the impact on performance when you vary the number of seed data points? How would completely random selection impact the final performance.\n* The current method may face challenges in scalability. The method is currently tested only on relatively small-scale toy problems like the Virtual Tools and PHYRE benchmarks. In real-world scenarios, physical systems are often high-dimensional, with more complex and nuanced dynamics. I also noticed a recently proposed multi-step challenge similar to PHYRE called I-PHYRE [1]. Would it be easy to apply the proposed method to the new challenge?\n\n[1] Li, Shiqian, et al. \"I-PHYRE: Interactive Physical Reasoning.\" The Twelfth International Conference on Learning Representations.", "other_comments_or_suggestions": "See above.", "questions_for_authors": "See above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Xj1lpvivzu", "reviewer": "Reviewer_d7h6", "summary": "The paper presents Causal-PIK, a novel method for causality-based physical reasoning that leverages a Physics-Informed Kernel within a Bayesian optimization framework. The primary focus is on single-intervention physical reasoning tasks, where an agent must make decisions based on the causal effects of its actions in complex environments.\n\nCausal-PIK incorporates causal reasoning into the decision-making process, helping agents efficiently explore and learn from their interactions with the environment. The findings demonstrate that Causal-PIK significantly outperforms state-of-the-art methods on benchmarks like Virtual Tools and PHYRE, achieving higher success rates while requiring fewer actions to solve tasks. In experiments, Causal-PIK achieved an AUCCESS score of 65.0 on the Virtual Tools benchmark, surpassing previous models by 7 points. On the PHYRE benchmark, it attained an AUCCESS score of 51.3, outperforming the best prior results by 9 points.\n\nThe algorithm iteratively updates a Gaussian process to model the scoring function based on previous actions and their outcomes. It selects actions that maximize an Upper Confidence Bound acquisition function, informed by the Physics-Informed Kernel, which reflects both causal effects and action similarities. Overall, Causal-PIK advances the field of physical reasoning in AI by integrating causal insights into the action selection process, demonstrating improved efficiency and effectiveness in solving complex reasoning tasks.\n\n## Update after rebuttal\n\nI appreciate the authors' response, and I will increase my score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer d7h6 for their thoughtful feedback. We are glad they found the method addresses an important aspect of physical reasoning, that is learning from feedback. We address the reviewer’s comments and will incorporate all of the following discussions in the final draft.\n\n> [d7h6.1] Resistance to noisy causal effect predictions \n\nWe appreciate the reviewer’s suggestion regarding the impact of the dynamics model’s accuracy on our method’s performance. To address this, we conducted an additional experiment to analyze the effect of the accuracy of the dynamics model.\n\nInstead of artificially adding more noise to existing predictions, we aimed to demonstrate that the ones we used were already highly noisy. To do this, we trained the dynamics model on tasks from the test templates, ensuring prior exposure to similar puzzles. As a result, the L2 error for object bounding boxes improved to 3.56, compared to 19.3 ± 4.55 when tested on entirely unseen puzzles.\n\nDespite this difference in prediction accuracy, our method achieved an AUCCESS of 45, which is only 4 points higher than the 41.6 ± 9.33 AUCCESS we reported for the case with unseen dynamics. This demonstrates that even with a substantial increase in prediction error, the performance drop is small, indicating that our method remains resilient to noisy dynamic predictions. While improved dynamic predictions can enhance performance, our approach does not rely on perfect predictions, retaining robustness even in the presence of inaccuracies.\n\n> [d7h6.2] Causal-PIK tested on I-PHYRE\n\nPlease refer to comment [vdey.2].\n\n> [d7h6.3] Why is the method based on causality\n\nWe appreciate the reviewer’s concern regarding our claims about causality and would like to clarify how our Physics-Informed Kernel explicitly captures causal relationships. We define causality as the dependence of an effect on its preceding causes. In our proposed method, this dependence is captured as follows:\n\n* Action-Effect Modeling\n\nUnlike models that rely purely on statistical correlations (e.g., our ablation with an RBF Kernel), our Physics-Informed Kernel is designed to encode causal dependencies between actions and their physical consequences. Instead of clustering actions based on feature-space proximity, our approach evaluates their actual impact on the system. For instance, two actions occurring in different spatial locations may be considered similar if they lead to the same physical outcome, such as maintaining the environment in a stationary state.\n\n* Causal Structure in the Reward Function\n\nOur method explicitly quantifies an action’s effect by normalizing the observed distance by the no-action baseline. This ensures that the kernel measures the true causal influence of an action rather than confounding factors.\n\n> [d7h6.4] Learning from feedback\n\nWe appreciate the reviewer's feedback regarding the technical contributions of our work. We believe our technical contributions extend significantly beyond introducing BO to maintain a belief over action proposals. A key innovation in our Physics-Informed Kernel is its ability to generalize knowledge across actions based on their causal effects. Unlike methods that learn only from direct observations, our approach leverages physical reasoning to infer the outcomes of untested actions that are predicted to share the same causal effect as observed ones. This means that from a single rollout, our model updates its belief not just about the executed action, but also about all actions that are predicted to produce a similar physical outcome. This significantly enhances sample efficiency and enables reasoning about alternative scenarios without exhaustive exploration.\n\nWe acknowledge that aspects of our presentation could be clarified, and we will refine the manuscript to better articulate these technical contributions in the final draft.\n\n> [d7h6.5] Improving Figures 2, 3, and 4\n\nWe appreciate the reviewer’s feedback on how to make Figures 2, 3, and 4 better. We will include these modifications in the final manuscript draft.\n\n> [d7h6.6] Updated PHYRE results for our method (Table 2)\n\nPlease refer to comment [jF9J.2].\n\n> [d7h6.7] Unclear writing with duplicate usage of words\n\nWe appreciate the reviewer’s feedback regarding unclear portions of the manuscript. We will work on improving the writing such as removing the duplicate usage of causal and action similarity in the final manuscript draft.\n\n> [d7h6.8] Extra evaluation metrics\n\nWe appreciate the reviewer’s suggestions and we will include a breakdown of the average success rate per puzzle and the average number of attempts per puzzle in the supplementary material.\n\n> [d7h6.9] Scaling to bigger states and actions and generalizing to more complex scenarios\n\nPlease refer to comments [jF9J.1] and [jF9J.3].", "claims_and_evidence": "The manuscript claims that the proposed method is based on causality. However, there is no clear definition of what causality it possesses or a piece of strong evidence to support that the model really captures causality that other models cannot.", "methods_and_evaluation_criteria": "The use of BO is reasonable to model the iteratively updated action proposal of solving a physical puzzle. The kernel design is grounded in physics and can be helpful to the next choice. The evaluation metric (AUCCESS) is from the original PHYRE paper and can reflect the model performance. However, the authors could consider including more diverse metrics that can more directly reflect the efficiency of solving the puzzles such as the attempts used to successfully solve a puzzle.", "theoretical_claims": "The theoretical formula including the iterative action proposal and the similarity design is correct.", "experimental_designs_or_analyses": "The authors validate their method on two physical reasoning tasks, with notable improvement compared with previous baselines. However, the two tasks both only consider single-step interventions. To further verify the proposed method, the authors could consider using physical reasoning tasks requiring multi-step interventions to solve (e.g. I-PHYRE). Besides, I think the authors could test against a more dynamic prediction model with BO to see if there is consistent improvement. What if the dynamic prediction is not accurate? Is there any analysis of how the prediction errors affect the performance of the proposed method?", "relation_to_broader_scientific_literature": "Previous related work addresses solving such physical reasoning tasks by either proposing better dynamic prediction models or by designing better action proposal methods. This work lies in the second school. Prior work utilizes RL to propose actions but falls short in terms of efficiency. This work tries to minimize this gap using a physics-informed kernel to update under the BO framework.", "essential_references_not_discussed": "No other reference should be discussed.", "other_strengths_and_weaknesses": "The manuscript addresses an important aspect of physical reasoning, that is learning from feedback. However, the main contribution of this work is to introduce BO to maintain the belief in action proposals. The technical soundness, thus, is a little bit weak. The experimental results and furtherThe writing is no clear enough. For example, there are many confused or duplicate usage of words such as causal similarity and action similarity.\n\nFigure 2 and figure 4 have too much blank area. Figure 3 is a little confusing to convey important information. analysis are also inadequate.", "other_comments_or_suggestions": "The writing is no clear enough. For example, there are many confused or duplicate usage of words such as causal similarity and action similarity.\n\nFigure 2 and figure 4 have too much blank area. Figure 3 is a little confusing to convey important information.", "questions_for_authors": "The authors should better explain what makes the proposed method a causality-based model. What about the generalization of the physics-informed kernel on other tasks?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: Causal-PIK proposes to tackle physical reasoning puzzles (specifically, Phyre and Visual Tools) by using a GP (with a "physics informed" kernel) as a surrogate for the dynamics and Bayesian Optimization to solve the search problem of which action to take in the environment to solve the puzzle.

All reviewers recommend acceptance. There are some remaining weak points with this paper:

- It overemphasizes the role of causality. Causality is even in the name of the proposed method, but it only considers a single initial intervention. The problem can be purely posed as search over this initial intervention, without appealing to causality. In fact, it is unlikely that this method would be useful in cases in which multiple interventions are necessary. Though the claim of causality is technically correct, this work doesn't involve any methods from the causality literature or solve any problems in the causality realm. Any method operating in a real (or simulated version of real) environment can be claimed as causal in the same sense this method is.
- This method will probably not scale without further modifications. This applies to multiple interventions or exponentially larger search spaces in the single intervention.

Despite these limitations, this work is still valuable in that it provides a new SOTA in two established benchmarks for physical reasoning. The first weak point is mostly in the writing and not in the method, whereas the second could be seen as an inherent limitation of the benchmarks themselves.
Decision: Accept (poster)