Physics-Informed MCTS

• Physics-Informed MCTS is an approach that enriches classical MCTS with embedded physical laws, constraints, and uncertainty modeling to guide optimal decision-making.
• It leverages methodologies like physics-informed neural networks, surrogate-guided rollouts, and GP-based uncertainty quantification to enhance exploration and ensure constraint satisfaction.
• Applications span robotics, scientific design, and building control where data scarcity, high simulation costs, and physical fidelity are critical for robust performance.

Physics-Informed Monte Carlo Tree Search (MCTS) integrates physical models, constraints, and surrogate-based guidance into the MCTS framework to achieve efficient, data-efficient, and constraint-respecting optimization or control in complex high-dimensional, continuous, or uncertain domains. This approach leverages either embedded physical equations (e.g., as regularizers in PINNs), uncertainty-aware surrogates, or hard-coded physical priors to inform decision-making at every stage of the tree search. Applications span scientific design, robotics, building control, and manipulation where data are limited, simulation costs are high, and reliable constraint satisfaction is critical.

1. Theoretical Foundations and Definition

Physics-Informed MCTS extends classical MCTS—an anytime, sample-based algorithm for optimal sequential decision problems—by incorporating physical knowledge into the search process. Physical information is introduced through one or more mechanisms:

• Embedding physical laws, invariants, or models into state transition or reward prediction, typically via physics-informed neural networks (PINNs), differential constraints, or learned surrogates with uncertainty quantification.
• Shaping the search tree and action selection via physics-derived priors, hard safety/pruning constraints (e.g., kinematic feasibility), or reward shaping that encodes constraint violation penalties.
• Integrating model uncertainty (epistemic or aleatoric) to bias exploration away from uncertain or untrustworthy branches.

The overarching goal is to enable robust, physically plausible, and data-efficient exploration and exploitation in settings where black-box evaluation is expensive, state/action spaces are large or continuous, and fidelity to physical laws is critical (Banik et al., 10 Jan 2026).

2. Algorithmic Components and Mathematical Formalism

Physics-Informed MCTS adheres to the canonical MCTS pipeline of Selection, Expansion, Simulation (Rollout), and Backpropagation, with domain-specific augmentations at each phase:

Selection:

Choose an action $a^*$ at state $s$ maximizing a composite score that incorporates exploitation, exploration, and physics-informed bias:

$$a^* = \arg\max_a \left[ Q(s, a) + C \sqrt{\frac{\ln N(s)}{N(s, a)}} + P_{\rm physics}(s, a) \right]$$

where $Q(s, a)$ is the mean value, $U(s, a)$ is the UCB term, and $P_{\rm physics}(s, a)$ represents domain-informed bonuses, such as constraint satisfaction or symmetry preservation (Banik et al., 10 Jan 2026).

Expansion:

Sample new child nodes via physics-aware distributions—using surrogate-guided direction/distance sampling, isotropic kernels, or action-space discretization informed by system constraints. For instance, in high-dimensional design, surrogate logistic regression models $\pi_{\rm sur}(a|s)$ bias sampling toward promising directions or radii according to learned local landscape information (Banik et al., 10 Jan 2026).

Simulation (Rollout):

Rather than relying solely on random or purely data-driven rollouts, employ PINN-based or physics-constrained models to autoregressively predict system evolution. Rollouts may be further bias-corrected via Gaussian Process UCB to account for model error,

$$r_{\rm sim} = r_{\rm PINN} + \mu_{\rm GP}(a) + \sqrt{\beta} \sigma_{\rm GP}(a)$$

as seen in manipulation and robot planning domains (Chopra et al., 2024).

Backpropagation:

Rewards are shaped to penalize constraint violations. Updates follow classical MCTS, but may account for additional penalties or uncertainties,

$$Q(s,a) \leftarrow Q(s,a) + \frac{1}{N(s,a)}(R - Q(s,a))$$

where $R(s,a)$ may include negative objectives, quadratic constraint penalties, and hard physical infeasibility (Banik et al., 10 Jan 2026).

Uncertainty-aware branch pruning:

Selection and expansion may be further adjusted to avoid high-uncertainty regions, discarding high-variance branches using softmax-scaled trust factors or sigmoid-based pruning (Faroni et al., 28 Jul 2025).

3. Representative Architectures

Physics-Informed Neural Networks (PINNs)

• PINNs serve as differentiable simulators, learning state transitions subject to collocated physical loss (enforcing ODEs or PDEs) and data loss. For example, trajectory skill models $F_\theta(x_0, t, \lambda)$ are trained with

$L(\theta) = L_D(\theta) + \epsilon L_P(\theta)$

where $L_D$ is MSE data loss and $L_P$ is physics residual loss. Latent system parameters (e.g., friction) can be treated as additional learnable variables, enabling online adaptation (Chopra et al., 2024).

Surrogate-Guided Rollouts

• Surrogates (logistic direction/distance models) are trained online to bias action sampling toward effective directions and scales, learned from observed local reward gradients. This mechanism focuses sampling and exploration effort (Banik et al., 10 Jan 2026).

Uncertainty Quantification and GP Correction

• GP models are trained to capture residual errors between fast PINN rollouts and high-fidelity evaluations. Uncertainty penalization ($-\lambda \sigma_{\text{GP}}(a)$) is incorporated directly in the UCT, and an adaptive threshold determines whether a PINN or real simulator is used (Vagadia et al., 2024, Faroni et al., 28 Jul 2025).

Hybrid Hard-Constraint Pruning

• State-dependent action set restriction is used (e.g., via a backup controller or feasibility classifier) to eliminate expansions guaranteed by physics to be invalid, increasing search efficiency (Pavirani et al., 2023, Zhu et al., 2022).

4. Applications Across Domains

Computational Design and Scientific Optimization

Physics-Informed MCTS operates in high-dimensional continuous spaces for challenging tasks such as crystal structure prediction, potential fitting, and engineering design. Integration of reward shaping, physics-informed priors, and local surrogate models enables efficient search over rugged, constrained landscapes. Benchmarks show superior or comparable convergence and robustness compared to global optimizers such as Particle Swarm or Whale Optimization Algorithm, and scalability to $d \sim 100$ (Banik et al., 10 Jan 2026).

Robot Manipulation and Physical Reasoning

PINN-augmented MCTS is used to compose dynamic skills (throw, slide, bounce, etc.) for 3D manipulation tasks. The approach enables few-shot learning of physical tasks, rapid adaptation to previously unseen environments or physical parameters, and improved data efficiency relative to model-free RL. In the PhyPlan framework, regret is reduced by 2–5$\times$ compared to model-free baselines, and planning speed is improved 4–6$\times$ via fast PINN rollouts (Chopra et al., 2024, Vagadia et al., 2024).

Building Climate Control

Physics-informed heating control is formulated as an MDP, with MCTS integrating a 2R2C-regularized PINN for temperature forecasts under continuous control. Compared to black-box simulators, PiNN-based MCTS reduces MAE by 32% with only 2 days of data, improves daily control reward by 3%, reduces energy cost by 4%, and reduces comfort-violation rates by 7% (Pavirani et al., 2023).

Liquid Handling and Deformable Object Planning

Uncertainty-aware MCTS penalizes actions whose model-predicted transitions fall in untrusted or poorly trained regions, as quantified by GP variance. This reduces open-loop planning failure rates, achieving 100% success even with sparse demonstrations, whereas standard MCTS collapses under model misspecification. Extensions generalize to folding, suction-cup insertion, and pushing tasks under sim-to-real uncertainty (Faroni et al., 28 Jul 2025).

5. Quantitative Performance and Comparison

Application Domain	Physics-Informed Model	Main Baselines Compared	Performance Metrics	Notable Results
High-Dimensional Optimization (Banik et al., 10 Jan 2026)	Surrogate+Physics-in-MCTS	WOA, PSO	Best objective, convergence rate, robustness	Top performance on 23 tasks, $d\leq 105$
Robot Skills (Chopra et al., 2024)	PINN + GP-UCB	DQN, PPO	Regret, skill data-efficiency, runtime	2–5$\times$ lower regret, 3–5$\times$ faster
Building Control (Pavirani et al., 2023)	2R2C PiNN + MCTS	Black-box NN, Bang-bang	$\Delta$MAE, daily reward, cost, comfort	$-32$\% MAE, $-$4\% cost, $+$7\% comfort
Liquid Pouring (Faroni et al., 28 Jul 2025)	GPR + Uncertainty MCTS	Standard/Inflated MCTS	Success rate, #actions, sensitivity	100\% success w/ $>$75\% data drop; robust to noise

These results demonstrate substantial improvements in sample complexity, constraint satisfaction, and computational efficiency across diverse domains.

6. Limitations, Open Problems, and Future Directions

Physics-Informed MCTS performance is fundamentally limited by the fidelity of the embedded physical model. PINN-based rollouts are only as accurate as their governing equations and the coverage of the training data; complex physical phenomena (e.g., friction, contacts, fluid dynamics) may remain challenging to represent compactly. Perception noise and sim-to-real gaps can degrade the reliability of planned actions. For hybrid approaches (such as PINN with GP-UCB correction), the quality of uncertainty estimation and the choice of switching thresholds are key factors in overall robustness (Chopra et al., 2024, Vagadia et al., 2024). Proofs of asymptotic optimality under nonstationary (uncertainty-penalized) selection rules remain incomplete, though empirical sample efficiency is strong (Faroni et al., 28 Jul 2025).

Potential improvements include direct end-to-end learning of skill-chaining, integration of language-based high-level reasoning to inform continuous parameterization, and extension to richer physical priors (deformable objects, fluids) and real-world deployment. Further, population-based and batch MCTS variants could further exploit parallelism in large-scale, high-dimensional problems (Banik et al., 10 Jan 2026).

7. Summary and Outlook

Physics-Informed MCTS establishes a paradigm in which tree search is guided and constrained by physical knowledge, achieving scalable, data-efficient, and constraint-satisfying decision-making in scientific computing, robotics, and control. By integrating PINN-based simulation, uncertainty-aware policy, domain-specific pruning, and surrogate guidance at all stages of the search, the approach enables reliable, interpretable optimization in complex environments where physical fidelity and efficiency are paramount. Ongoing advancements are likely to extend its applicability to broader physical domains and incorporate tighter integration with state estimation, perceptual modules, and high-level reasoning systems (Pavirani et al., 2023, Chopra et al., 2024, Faroni et al., 28 Jul 2025, Banik et al., 10 Jan 2026, Vagadia et al., 2024, Zhu et al., 2022).