Teleodynamic Learning in Adaptive Systems

• Teleodynamic learning is defined as the emergence of adaptive behavior through the coupled evolution of system structure, parameters, and endogenous resources.
• It integrates embodied, physical, and computational frameworks to autonomously generate dynamic attractors such as limit cycles and policy regimes.
• This paradigm shifts intelligence from static optimization to regulated reorganization, offering practical insights into multi-timescale learning and agent interactions.

Searching arXiv for recent and directly relevant papers on teleodynamic learning and adjacent formulations. Teleodynamic learning denotes a family of approaches in which learning is treated as the emergence and stabilization of functional organization in coupled dynamical systems rather than solely as the minimization of a fixed scalar objective. In the current literature, the relevant state variables differ by domain but repeatedly include structure, parameters, endogenous resources, morphology, bodily dynamics, memory systems, or interaction couplings; correspondingly, the learned target is often a dynamical attractor—such as a stable policy regime, a morphology-conditioned behavioral repertoire, a trajectory, a sequence, or a limit cycle—rather than merely a static input–output map (Horst et al., 11 Mar 2026, Mandal et al., 2024, Gupta et al., 2021).

1. Conceptual foundations

A central formulation defines Teleodynamic Learning as a paradigm in which learning is modeled as “navigation of a coupled dynamical system” over three co-evolving quantities: the structure of the hypothesis class, the parameters within that structure, and an endogenous resource variable. In that formulation, intelligence is not identified with optimization over a fixed model family, but with the coupled evolution of what a system can represent, how it adapts, and which changes its internal resources can sustain. The stated biological motivation is teleodynamics in the sense of end-directed behavior emerging from “closure of constraints” and the mutual maintenance of processes and structures (Horst et al., 11 Mar 2026).

A second formulation arises in physical learning of dynamical behaviors. There, the goal is explicitly time-dependent: a material or active system acquires the ability to autonomously generate a target trajectory, pathway, or limit cycle after exposure to examples. The learned behavior is encoded not as a static minimum but “in limit cycles or pathways of a dynamical system,” and retrieval is the system’s autonomous evolution toward that behavior after training has ended. This formulation makes teleodynamic learning inseparable from non-equilibrium dynamics, causal propagation, and attractor selection (Mandal et al., 2024).

A third formulation is embodied and evolutionary. In Deep Evolutionary Reinforcement Learning, teleodynamic organization is distributed across three coupled timescales: slow evolutionary dynamics over genotypes, intermediate morphological dynamics linking genotype to body and physical regime, and fast within-lifetime reinforcement learning and control. Evolution does not directly optimize behavior; it selects morphologies that allow reinforcement learning to achieve high reward more effectively in a given niche. This places learnability itself under evolutionary pressure (Gupta et al., 2021).

Across these formulations, a recurring claim is that teleodynamic learning does not localize intelligence in a single substrate. The controller, the body, the energy landscape, the replay system, or the interaction topology may each carry part of the adaptive burden. This suggests that “learning” in teleodynamic settings is best understood as regulated reorganization of a whole coupled system rather than as parameter fitting in isolation.

2. Formal structure and characteristic dynamics

The most explicit machine-learning formalization defines a Teleodynamic Learning System as a tuple

$\mathcal{T} = (S, X, Y, A, \gamma, J, E_0),$

with state space

$$S = H^* \times M \times \mathbb{R}^+ \times (X \times Y)^* \times \text{Registry}.$$

Here the state contains a structured hypothesis set $\mathcal{H}$, parameters $\theta$ on a statistical manifold, an endogenous resource $E$, a history $\tau$, and a registry for reusable compressed subforms. Inner dynamics perform natural-gradient parameter updates,

$$\theta^{(t+1)} = \theta^{(t)} - \eta\,F(\theta^{(t)})^{-1}\nabla_\theta L(\theta^{(t)};x^{(t)},y^{(t)}),$$

while outer dynamics choose discrete structural actions by minimizing a local teleodynamic objective

$$J(s,a,x,y)=L(s',x,y)+\lambda_C\,\Delta C+\lambda_E\,c_a,$$

subject to the resource constraint $\mathrm{cost}(a)\le E$. Resource is then updated by

$$E^{(t+1)}=\gamma_E E^{(t)}+\delta E^{(t)}-c_{a^{(t)}}.$$

This construction is explicitly contrasted with a monolithic objective of the form $$S = H^* \times M \times \mathbb{R}^+ \times (X \times Y)^* \times \text{Registry}.$$0, and is presented as yielding emergent structural halt, phase-structured learning dynamics, and convergence guarantees grounded in information geometry rather than convexity (Horst et al., 11 Mar 2026).

A related but physically grounded formalism models learning as coupled state–interaction dynamics. In the modified Hopfield system for dynamic behaviors, spins evolve as

$$S = H^* \times M \times \mathbb{R}^+ \times (X \times Y)^* \times \text{Registry}.$$1

while the couplings themselves obey a delayed STDP-like rule

$$S = H^* \times M \times \mathbb{R}^+ \times (X \times Y)^* \times \text{Registry}.$$2

The paper’s central result is that two ingredients are necessary to learn time-dependent behaviors: learning rules with time delays and training examples that break time-reversal symmetry. With $$S = H^* \times M \times \mathbb{R}^+ \times (X \times Y)^* \times \text{Registry}.$$3, antisymmetric couplings decay away; with $$S = H^* \times M \times \mathbb{R}^+ \times (X \times Y)^* \times \text{Registry}.$$4 and temporally directed training, non-reciprocal couplings can accumulate and support limit-cycle retrieval (Mandal et al., 2024).

Energy-based dynamical models provide a third mathematical template. An energy-based system is defined by

$$S = H^* \times M \times \mathbb{R}^+ \times (X \times Y)^* \times \text{Registry}.$$5

together with a scalar energy $$S = H^* \times M \times \mathbb{R}^+ \times (X \times Y)^* \times \text{Registry}.$$6 satisfying

$$S = H^* \times M \times \mathbb{R}^+ \times (X \times Y)^* \times \text{Registry}.$$7

The canonical case is gradient flow,

$$S = H^* \times M \times \mathbb{R}^+ \times (X \times Y)^* \times \text{Registry}.$$8

with fast inference/retrieval occurring at fixed $$S = H^* \times M \times \mathbb{R}^+ \times (X \times Y)^* \times \text{Registry}.$$9 and slower learning reshaping the landscape through $\mathcal{H}$0. In this setting, attractors are the computation, and learning is the slow sculpting of the energy landscape explored by the fast state dynamics (Montanari et al., 6 Apr 2026).

These formalisms differ in ontology, but all replace a single external optimizer with coupled processes whose relative timescales matter. Teleodynamic learning therefore typically appears where the distinction between “model,” “memory,” “controller,” and “substrate” becomes dynamical rather than architectural.

3. Embodied, physical, and material realizations

Embodied evolution provides one of the clearest teleodynamic case studies. In DERL, a population of morphologies evolves asynchronously by tournament selection and mutation, but controllers are always reinitialized tabula rasa and trained from scratch with PPO. Morphology alone is inherited. The genotype is a kinematic tree with a spherical head, cylindrical limbs, and motorized hinge joints; mutations alter growth, deletion, density, joint degrees of freedom, limits, and gear. The resulting phenotype is a physically simulated body whose passive dynamics and control interfaces shape the reinforcement-learning landscape. On this basis, the authors define “morphological intelligence” as how much a morphology facilitates fast and high-performance learning of a suite of novel tasks. Their evaluation shows that morphologies evolved in complex niches learn faster and transfer better: at full training budget, MVT morphologies outperform FT in 7/8 tasks, and at reduced budget the VT/MVT advantage becomes more pronounced across all tasks. Over roughly ten generations, iterations-to-criterion decrease by about a factor of 2, which the paper identifies as the first demonstration of a morphological Baldwin effect; physically, this is associated with increased passive stability and decreased cost of work (Gupta et al., 2021).

Physical learning of dynamical behaviors pushes the same logic into material and active-matter systems. In LEGO experiments, two motorized “spins” are trained by recording angle time series and updating couplings with a causal temporal kernel; symmetric training yields reciprocal couplings and fixed-point retrieval, whereas training with a phase lag yields asymmetric couplings and autonomous limit-cycle motion. In chemotactic particle simulations, particles emit and sense fields, and the effective learning rule for mobilities emerges from causal Green’s-function propagation: $\mathcal{H}$1 Because this rule is history-dependent and time-asymmetric when motion generates a wake, non-reciprocal couplings can be learned without any externally programmed learning code. The reported behaviors include rotating ring self-assembly, run-and-chase dynamics, and flying-wing target capture (Mandal et al., 2024).

Robotic co-adaptation supplies an additional embodied variant. A tendon-driven, over-actuated, backdrivable biped with two joints and three actuators per leg uses a simple 3-layer neural network to learn an inverse map from 6D joint kinematics to 3D motor activations after only 2 minutes of motor babbling. “Natural” babbling uses sinusoidal, phase-structured, muscle-like actuation and yields consistent cyclical movements in air; without further tuning, those cycles produce locomotion when the biped is lowered into slight ground contact. “Naive” babbling, which persistently coactivates antagonist actuators, does not produce consistent locomotion under slight contact, but locomotion emerges when the desired leg trajectories are moved 1 cm below ground, making desired–obtained error unavoidable. The paper’s explicit conclusion is that locomotion can emerge from brain–body–environment interactions “without explicit control of trajectory errors” and that stronger environmental interference can improve locomotor success by reducing feasible configuration space (Urbina-Meléndez et al., 2024).

Taken together, these results treat body and environment as active components of the learning dynamics. A plausible implication is that teleodynamic learning is most naturally realized when adaptation is partly delegated to physics: morphology, compliance, contact, and passive stability can regularize and canalize learning in ways that controller-only formalisms do not capture.

4. Dynamical, neural, and multi-agent computational realizations

Several computational lines instantiate teleodynamic learning without explicit embodiment. Fixed-weight recurrent networks offer one version. After a pretraining phase that sculpts reservoir dynamics and associates output behaviors with low-dimensional context variables, the network of Klos, Iliescu, and Lazar can dynamically learn new target dynamics by a short exposure to an error signal, while all weights remain fixed. The state equations

$\mathcal{H}$2

support fast acquisition of new oscillatory, driven, and chaotic dynamics as state-space reconfiguration rather than synaptic modification. The paper explicitly interprets this as learning realized “in the ongoing state–trajectory of a system” (Klos et al., 2019).

Spiking dynamical networks provide a second version. In non-negative sparse coding and dictionary learning, equilibria of a leaky integrate-and-fire network correspond to solutions of the $\mathcal{H}$3-regularized sparse coding subproblem, while a two-phase contrastive protocol with top-down feedback yields local gradient estimates for dictionary updates. The desired objective

$\mathcal{H}$4

is not solved by an external optimizer; inference and gradient estimation are implemented by the network’s own trajectory and by differences between two limiting states. This makes learning signals local and dynamical rather than backpropagated through an external computation graph (Lin et al., 2018).

Energy-based models generalize attractor-based computation across memory retrieval, sampling, optimization, and constrained reconstruction. Continuous-time Hopfield networks, Boltzmann machines, dense associative memories, oscillator-based networks, and proximal-gradient flows are all treated as systems whose trajectories descend or sample explicit energy landscapes. In this literature, fixed points, phase-locked states, and stochastic stationary distributions become the operative computational objects, and learning is the slow reshaping of the underlying energy function (Montanari et al., 6 Apr 2026).

Teleodynamic structure also appears in continual and meta-learning. The task-agnostic continual meta-learning framework of Caccia et al. separates “what task is being solved” from “how the task should be solved,” with fast task-specific adaptation nested inside slower continual updates of task-agnostic meta-parameters. CLS-ER similarly organizes continual learning into interacting episodic memory, short-term semantic memory, and long-term semantic memory; the working model is constrained by replay and by semantic consistency, while plastic and stable semantic models are maintained as EMA processes on distinct timescales (He et al., 2019, Arani et al., 2022).

In multi-agent systems, Learning with Opponent-Learning Awareness extends teleodynamic reasoning to strategic interaction. Each agent updates through the anticipated next learning step of the opponent,

$\mathcal{H}$5

so the learning rule explicitly shapes future learning dynamics rather than treating the opponent as a static environment. The reported emergence of tit-for-tat-like cooperation in the iterated prisoner’s dilemma and convergence to Nash behavior in repeated matching pennies show that teleodynamic organization can also be distributed across coupled learners (Foerster et al., 2017).

5. Recurring empirical phenomena and common misconceptions

Across these implementations, several empirical regularities recur. One is self-stabilization. In the Distinction Engine DE11, a structural noop action is always available, and structural exploration halts when noop becomes locally optimal under the teleodynamic objective; the system thereby enters a mature regime with fixed structure but continuing parametric learning. The paper presents this as “emergent structural halt” rather than external early stopping (Horst et al., 11 Mar 2026).

A second regularity is phase structure. Teleodynamic trajectories are repeatedly described as moving through distinguishable regimes. DE11 exhibits under-structuring, teleodynamic growth, and equilibrium or over-structuring, measurable through structural transition rates and trajectories in complexity–energy space. In embodied evolution, early morphologies require more of the lifetime to acquire competent behavior, while later morphologies learn substantially earlier in life. In bipedal co-adaptation, slight ground contact destabilizes some cycles, but stronger constraint can restabilize them by narrowing feasible dynamics (Horst et al., 11 Mar 2026, Gupta et al., 2021, Urbina-Meléndez et al., 2024).

A third regularity is that useful learning often depends on asymmetry, irreversibility, or context. In physical learning of dynamical behaviors, time-delayed learning rules are insufficient unless training itself breaks time-reversal symmetry; only then do non-reciprocal couplings and dynamic attractors emerge. In continual meta-learning, the emphasis shifts from preserving zero-shot performance on old tasks to “faster remembering,” namely rapid recovery once a small amount of context from the old task is reintroduced. In CLS-ER, recent and long-term semantic memories play complementary roles precisely because a single timescale does not suffice for both rapid adaptation and durable retention (Mandal et al., 2024, He et al., 2019, Arani et al., 2022).

These results also clarify several misconceptions. Teleodynamic learning does not require the absence of objectives; rather, in some formulations objectives become local, myopic, or physically embodied instead of globally fixed. Nor does it require that learning be weightless or structureless: some systems rely on slow structural change, others on fixed weights with state-space reconfiguration, and still others on simultaneous state and coupling dynamics. Likewise, the relevant “goal” need not be a final static configuration. In many of the cited systems, the learned target is explicitly a trajectory, a sequence, a limit cycle, or a morphology-conditioned regime of behavior.

6. Limits, unresolved questions, and research directions

The literature also makes clear that teleodynamic learning remains a heterogeneous research program rather than a fully unified theory. The most explicit ML formalization is still limited by scalability, diagonal Fisher approximations, sensitivity to $\mathcal{H}$6, $\mathcal{H}$7, and $\mathcal{H}$8, a limited structural repertoire (genesis, wedge, noop), and an energy variable whose semantics are acknowledged to be conceptual rather than physically grounded. On standard benchmarks, DE11 is competitive on small tabular datasets but described as “underpowered” for DIGITS, which indicates the current limits of its representational substrate (Horst et al., 11 Mar 2026).

Physical and embodied versions face different constraints. DERL is simulation-only, omits ontogenetic development, evolves morphology but not controllers, and uses externally specified reward functions rather than intrinsic goals. The physical learning of dynamical behaviors paper identifies delayed learning and time-reversal-symmetry breaking as necessary conditions, but more complex teleodynamic systems would require richer chemical, mechanical, or multi-species embodiments. The dynamical invariants framework goes further by proposing a thermodynamic basis for autonomous learning—predictable information as an energy source quantified by $\mathcal{H}$9 per correctly predicted bit—but leaves controller design, hardware–software interplay, and practical scale as open problems (Gupta et al., 2021, Mandal et al., 2024, Ushveridze, 2024).

Several research directions follow directly from these limitations. One is to couple development, morphology, and neural adaptation in a single framework, rather than isolating only one adaptive substrate. Another is to replace externally specified reward or label signals with endogenous constraints derived from resource maintenance, information flow, or homeostatic regulation. A third is to extend teleodynamic learning from small-scale interpretable systems and stylized physical media to high-dimensional, open-ended environments without losing the distinctive features of multi-timescale coupling, self-stabilization, and attractor-based organization. A plausible implication is that the long-term significance of teleodynamic learning will depend on whether these distinct strands—resource-coupled ML, embodied evolution, physical dynamical learning, energy-based computation, and continual memory systems—can be brought into a common formal vocabulary without erasing their domain-specific mechanisms.

In its present state, teleodynamic learning names a convergent research tendency: learning is increasingly modeled as an organized dynamical process that maintains and reshapes its own conditions of success. Whether the substrate is a logical rule engine, an evolving morphology, a spiking circuit, a recurrent reservoir, an active material, or a replay-driven continual learner, the shared theme is that adaptive competence is not merely optimized from outside. It is stabilized from within.