Inverse Problems in Soft Matter and Soft Robotics

How do we find configurations that optimize something? And why are inverse problems so much harder than forward ones?

What are we optimizing?

In any physical system, the forward problem asks: given the current state and the equations of motion, what happens next? Forward problems are often hard (nonlinear, multi-scale, stochastic), but they are at least conceptually clear — just evolve the system.

The inverse problem asks instead: given a desired outcome, what inputs or configurations produce it? The “something” being optimized might be:

Free energy (the system settles into its thermodynamic ground state)
Entanglement (maximize the topological constraint between components)
Control effort (minimize the energy needed to steer the system to a target)
Target shape (minimize the distance from the current configuration to a desired shape)
Mechanical response (achieve a target stiffness or compliance)

Each of these gives a different optimization problem, but they share a common structure: a high-dimensional configuration space, a scalar objective, and (usually) nonlinear constraints.

Forward vs. inverse: a fundamental asymmetry

Forward problems in soft matter are already difficult. Simulating a dense rod packing, computing the elastic equilibrium of a network, or predicting the trajectory of a hyper-redundant robot all require careful numerics and significant computation.

Why are inverse problems so much harder?

The core reason is non-uniqueness and non-convexity. For most forward problems, the state at time \(t+dt\) is uniquely determined by the state at time \(t\) (given the equations of motion). But for inverse problems, many configurations may achieve the same objective — or the landscape may have many local optima that are easy to find but far from the global optimum. Gradient descent finds a local solution, not necessarily the best one.

Additionally, inverse problems often involve implicit constraints that are hard to differentiate through (contact, topology, inextensibility), and the forward model may be expensive to evaluate, limiting the number of gradient steps we can take.

Examples from my work

Snake control: hyper-redundant inverse kinematics

A snake-like robot with \(N\) joints has \(N\) degrees of freedom, but the task (e.g., “grasp this object”) may only require controlling a few end-effector coordinates. The extra DOFs are redundant — there is a whole manifold of solutions. The question is: which solution on this manifold is best (safest, most energy-efficient, most robust)?

In snake-control, I use the Gauss linking integral as an objective to guide the robot body to topologically grasp a target — achieving a form of robust grasping that doesn’t require precise positioning.

Snake lattice: inverse design of networks

A network of elastic rods (a lattice) has a mechanical response determined by its topology and rest state. Given a desired response (e.g., “deform in this specific pattern when loaded”), find the rest state and connectivity that achieves it. This is an inverse design problem with topological degrees of freedom.

snake-lattice explores this by differentiating through the elastic energy of a curve network.

Rod packing: maximally entangled design

Given a set of rods in a box, can we find an arrangement that maximizes total entanglement? This is an inverse design problem where the “target” is not a shape but a topological property. The answer requires searching over the space of all valid (non-overlapping) configurations.

entanglement-optimization addresses this using gradient ascent on the linking number sum.

Adjoint methods and differentiable physics

The most powerful tool for solving inverse problems in physics is the adjoint method. Instead of computing gradients of the objective with respect to every parameter (which requires as many forward solves as there are parameters), the adjoint method computes the gradient using just two solves: one forward and one “adjoint” (a modified backward solve).

For time-dependent problems, the adjoint method is essentially backpropagation through time — the same algorithm used to train recurrent neural networks. In the physics context, this is called differentiable physics or differentiable simulation.

The emergence of JAX and similar frameworks (PyTorch, Jax, Enzyme) has made differentiable physics much more accessible. In my work, JAX is used in:

entanglement-optimization — differentiating through the Gauss integral
snake-control — differentiating through forward kinematics and topology
der-python — differentiating through elastic rod energies

Open questions

The field of differentiable physics for soft matter is rapidly evolving. Some open questions I find particularly interesting:

Contact and friction: These involve non-smooth events (stick-slip transitions, impact) that are hard to differentiate through. Smooth approximations exist but introduce bias.
Topology changes: When rods cross (in a simulation that allows it) or entanglement changes, the objective has discontinuities. How do we handle these?
Scalability: Gradient-based methods work well for \(N \sim 10^2\) rods. Can they scale to \(N \sim 10^4\) or \(10^6\)?
Integration with data-driven methods: Can differentiable simulators be combined with learned models to solve inverse problems more efficiently?

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