Causal Field Theory: Causal Semantics for PDE-Based Spatio-Temporal Systems

Arash Mehrjou, Bernhard Schölkopf

Keywords: causal inference, partial differential equations, spatio-temporal dynamics, response kernels, causal flux, intervention design, neural operators, physics-informed learning, reaction-diffusion systems, Green's functions, mechanistic interpretability

TL;DR: We extend causal semantics to PDE-based spatio-temporal systems via response kernels and causal flux, with natural connections to neural operators and physics-informed learning.

Abstract: Partial differential equations model complex spatio-temporal phenomena from fluid dynamics to biological systems, yet conventional PDE solvers do not typically offer causal semantics for interventions. We introduce Causal Field Theory (CFT), a framework that endows PDE-based systems with explicit causal structure through regional mechanism interventions, which modify evolution laws within spatial regions, and causal response kernels that quantify how influence propagates. CFT bridges mechanistic simulation and causal reasoning, enabling quantitative intervention comparison via causal flux and causal cones. We prove that macro-scale interventions aligned with coherent propagation modes generate greater causal flux than fragmented micro-scale perturbations; biological systems such as tissue-scale drug delivery and morphogen dynamics provide suitable examples of this principle. Experiments on reaction-diffusion PDEs illustrate the framework and its potential relevance for neural operators and physics-informed learning. CFT may enable interpretable causal analysis of PDE dynamics and principled intervention design for scientific AI.

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Submission Number: 14