TargetForms.cpp

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#include <polyfem/optimization/forms/TargetForms.hpp>
 
#include <polyfem/varforms/diff/DifferentiableVarForm.hpp>
#include <polyfem/Common.hpp>
#include <polyfem/io/Evaluator.hpp>
#include <polyfem/io/OBJWriter.hpp>
#include <polyfem/utils/Logger.hpp>
#include <polyfem/utils/Types.hpp>
#include <polyfem/utils/MaybeParallelFor.hpp>
#include <polyfem/utils/BoundarySampler.hpp>
#include <polyfem/assembler/Mass.hpp>
#include <polyfem/optimization/DiffCache.hpp>
 
#include <polyfem/utils/IntegrableFunctional.hpp>
 
#include <Eigen/Core>
#include <spdlog/fmt/fmt.h>
 
#include <algorithm>
#include <cassert>
#include <cmath>
#include <cstddef>
#include <map>
#include <memory>
#include <set>
#include <stdexcept>
#include <string>
#include <utility>
#include <vector>
 
using namespace polyfem::utils;
 
namespace polyfem::solver
{
    namespace
    {
        class LocalThreadScalarStorage
        {
        public:
            double val;
            assembler::ElementAssemblyValues vals;
 
            LocalThreadScalarStorage()
            {
                val = 0;
            }
        };
    } // namespace
 
    IntegrableFunctional TargetForm::get_integral_functional() const
    {
        IntegrableFunctional j;
        if (target_varform_)
        {
            assert(target_diff_cache_->size() > 0);
 
            auto j_func = [this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
                int e_ref = params.elem;
                if (auto search = e_to_ref_e_.find(params.elem); search != e_to_ref_e_.end())
                    e_ref = search->second;
 
                Eigen::MatrixXd pts_ref;
                target_varform_->primary_space().geometry_basis_list()[e_ref].eval_geom_mapping(local_pts, pts_ref);
 
                Eigen::MatrixXd u_ref, grad_u_ref;
                const Eigen::VectorXd &sol_ref = target_diff_cache_->u(target_varform_->get_problem().is_time_dependent() ? params.step : 0);
                io::Evaluator::interpolate_at_local_vals(target_varform_->get_mesh(), target_varform_->get_problem().is_scalar(), target_varform_->primary_space().basis_list(), target_varform_->primary_space().geometry_basis_list(), e_ref, local_pts, sol_ref, u_ref, grad_u_ref);
 
                val = (u_ref + pts_ref - u - pts).rowwise().squaredNorm();
            };
 
            auto djdu_func = [this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
                int e_ref = params.elem;
                if (auto search = e_to_ref_e_.find(params.elem); search != e_to_ref_e_.end())
                    e_ref = search->second;
 
                Eigen::MatrixXd pts_ref;
                target_varform_->primary_space().geometry_basis_list()[e_ref].eval_geom_mapping(local_pts, pts_ref);
 
                Eigen::MatrixXd u_ref, grad_u_ref;
                const Eigen::VectorXd &sol_ref = target_diff_cache_->u(target_varform_->get_problem().is_time_dependent() ? params.step : 0);
                io::Evaluator::interpolate_at_local_vals(target_varform_->get_mesh(), target_varform_->get_problem().is_scalar(), target_varform_->primary_space().basis_list(), target_varform_->primary_space().geometry_basis_list(), e_ref, local_pts, sol_ref, u_ref, grad_u_ref);
 
                val = 2 * (u + pts - u_ref - pts_ref);
            };
 
            j.set_j(j_func);
            j.set_dj_du(djdu_func);
            j.set_dj_dx(djdu_func); // only used for shape derivative
        }
        else if (have_target_func)
        {
            auto j_func = [this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
                val.setZero(u.rows(), 1);
 
                const Eigen::MatrixXd X = u + pts;
                for (int q = 0; q < u.rows(); q++)
                    val(q) = target_func(X(q, 0), X(q, 1), X.cols() == 2 ? 0 : X(q, 2), 0, params.elem);
            };
 
            auto djdu_func = [this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
                val.setZero(u.rows(), u.cols());
 
                const Eigen::MatrixXd X = u + pts;
                for (int q = 0; q < u.rows(); q++)
                    for (int d = 0; d < val.cols(); d++)
                        val(q, d) = target_func_grad[d](X(q, 0), X(q, 1), X.cols() == 2 ? 0 : X(q, 2), 0, params.elem);
            };
 
            j.set_j(j_func);
            j.set_dj_du(djdu_func);
            j.set_dj_dx(djdu_func); // only used for shape derivative
        }
        else // error wrt. a constant displacement
        {
            if (target_disp.size() == varform_->get_mesh().dimension())
            {
                auto j_func = [this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
                    val.setZero(u.rows(), 1);
 
                    for (int q = 0; q < u.rows(); q++)
                    {
                        Eigen::VectorXd err = u.row(q) - this->target_disp.transpose();
                        for (int d = 0; d < active_dimension_mask.size(); d++)
                            if (!active_dimension_mask[d])
                                err(d) = 0;
                        val(q) = err.squaredNorm();
                    }
                };
                auto djdu_func = [this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
                    val.setZero(u.rows(), u.cols());
 
                    for (int q = 0; q < u.rows(); q++)
                    {
                        Eigen::VectorXd err = u.row(q) - this->target_disp.transpose();
                        for (int d = 0; d < active_dimension_mask.size(); d++)
                            if (!active_dimension_mask[d])
                                err(d) = 0;
                        val.row(q) = 2 * err;
                    }
                };
 
                j.set_j(j_func);
                j.set_dj_du(djdu_func);
            }
            else
                log_and_throw_adjoint_error("[{}] Only constant target displacement is supported!", name());
        }
 
        return j;
    }
 
    void TargetForm::set_reference(const std::shared_ptr<const varform::DifferentiableVarForm> &target_varform, std::shared_ptr<const DiffCache> target_diff_cache, const std::set<int> &reference_cached_body_ids)
    {
        target_varform_ = target_varform;
        target_diff_cache_ = std::move(target_diff_cache);
 
        std::map<int, std::vector<int>> ref_interested_body_id_to_e;
        int ref_count = 0;
        for (int e = 0; e < target_varform_->primary_space().basis_list().size(); ++e)
        {
            int body_id = target_varform_->get_mesh().get_body_id(e);
            if (reference_cached_body_ids.size() > 0 && reference_cached_body_ids.count(body_id) == 0)
                continue;
            if (ref_interested_body_id_to_e.find(body_id) != ref_interested_body_id_to_e.end())
                ref_interested_body_id_to_e[body_id].push_back(e);
            else
                ref_interested_body_id_to_e[body_id] = {e};
            ref_count++;
        }
 
        std::map<int, std::vector<int>> interested_body_id_to_e;
        int count = 0;
        for (int e = 0; e < varform_->primary_space().basis_list().size(); ++e)
        {
            int body_id = varform_->get_mesh().get_body_id(e);
            if (reference_cached_body_ids.size() > 0 && reference_cached_body_ids.count(body_id) == 0)
                continue;
            if (interested_body_id_to_e.find(body_id) != interested_body_id_to_e.end())
                interested_body_id_to_e[body_id].push_back(e);
            else
                interested_body_id_to_e[body_id] = {e};
            count++;
        }
 
        if (count != ref_count)
            adjoint_logger().error("[{}] Number of interested elements in the reference and optimization examples do not match! {} {}", name(), count, ref_count);
        else
            adjoint_logger().trace("[{}] Found {} matching elements.", name(), count);
 
        for (const auto &kv : interested_body_id_to_e)
        {
            for (int i = 0; i < kv.second.size(); ++i)
            {
                e_to_ref_e_[kv.second[i]] = ref_interested_body_id_to_e[kv.first][i];
            }
        }
    }
 
    void TargetForm::set_reference(const json &func, const json &grad_func)
    {
        target_func.init(func, resources_);
        for (size_t k = 0; k < grad_func.size(); k++)
            target_func_grad[k].init(grad_func[k], resources_);
        have_target_func = true;
    }
 
    void SDFTargetForm::solution_changed_step(const int time_step, const Eigen::VectorXd &x)
    {
        const auto &bases = varform_->primary_space().basis_list();
        const auto &gbases = varform_->primary_space().geometry_basis_list();
        const int actual_dim = varform_->get_problem().is_scalar() ? 1 : dim;
 
        auto storage = utils::create_thread_storage(LocalThreadScalarStorage());
        utils::maybe_parallel_for(varform_->boundary_state().total_local_boundary.size(), [&](int start, int end, int thread_id) {
            LocalThreadScalarStorage &local_storage = utils::get_local_thread_storage(storage, thread_id);
 
            Eigen::MatrixXd uv, samples, gtmp;
            Eigen::MatrixXd points, normal;
            Eigen::VectorXd weights;
 
            Eigen::MatrixXd u, grad_u;
 
            for (int lb_id = start; lb_id < end; ++lb_id)
            {
                const auto &lb = varform_->boundary_state().total_local_boundary[lb_id];
                const int e = lb.element_id();
 
                for (int i = 0; i < lb.size(); i++)
                {
                    const int global_primitive_id = lb.global_primitive_id(i);
                    if (ids_.size() != 0 && ids_.find(varform_->get_mesh().get_boundary_id(global_primitive_id)) == ids_.end())
                        continue;
 
                    utils::BoundarySampler::boundary_quadrature(lb, varform_->n_boundary_samples(), varform_->get_mesh(), i, false, uv, points, normal, weights);
 
                    assembler::ElementAssemblyValues &vals = local_storage.vals;
                    vals.compute(e, varform_->get_mesh().is_volume(), points, bases[e], gbases[e]);
                    io::Evaluator::interpolate_at_local_vals(e, dim, actual_dim, vals, diff_cache_->u(time_step), u, grad_u);
 
                    normal = normal * vals.jac_it[0]; // assuming linear geometry
 
                    for (int q = 0; q < u.rows(); q++)
                        interpolation_fn->cache_grid([this](const Eigen::MatrixXd &point, double &distance) { compute_distance(point, distance); }, vals.val.row(q) + u.row(q));
                }
            }
        });
    }
 
    void SDFTargetForm::set_bspline_target(const Eigen::MatrixXd &control_points, const Eigen::VectorXd &knots, const double delta)
    {
        dim = control_points.cols();
        delta_ = delta;
        if ((dim != 2) || (varform_->get_mesh().dimension() != 2))
            log_and_throw_error("SDFTargetForm specified for 2d.");
 
        samples = 100;
 
        nanospline::BSpline<double, 2, 3> curve;
        curve.set_control_points(control_points);
        curve.set_knots(knots);
 
        t_or_uv_sampling = Eigen::VectorXd::LinSpaced(samples, 0, 1);
        point_sampling.setZero(samples, 2);
        for (int i = 0; i < t_or_uv_sampling.size(); ++i)
            point_sampling.row(i) = curve.evaluate(t_or_uv_sampling(i));
 
        {
            Eigen::MatrixXi edges(samples - 1, 2);
            edges.col(0) = Eigen::VectorXi::LinSpaced(samples - 1, 0, samples - 2);
            edges.col(1) = Eigen::VectorXi::LinSpaced(samples - 1, 1, samples - 1);
            io::OBJWriter::write(fmt::format("spline_target_{:d}.obj", rand() % 100), point_sampling, edges);
        }
 
        interpolation_fn = std::make_unique<LazyCubicInterpolator>(dim, delta_);
    }
 
    void SDFTargetForm::set_bspline_target(const Eigen::MatrixXd &control_points, const Eigen::VectorXd &knots_u, const Eigen::VectorXd &knots_v, const double delta)
    {
 
        dim = control_points.cols();
        delta_ = delta;
        if ((dim != 3) || (varform_->get_mesh().dimension() != 3))
            log_and_throw_error("SDFTargetForm specified for 3d.");
 
        samples = 100;
 
        nanospline::BSplinePatch<double, 3, 3, 3> patch;
        patch.set_control_grid(control_points);
        patch.set_knots_u(knots_u);
        patch.set_knots_v(knots_v);
        patch.initialize();
 
        t_or_uv_sampling.resize(samples * samples, 2);
        for (int i = 0; i < samples; ++i)
        {
            t_or_uv_sampling.block(i * samples, 0, samples, 1) = Eigen::VectorXd::LinSpaced(samples, 0, 1);
            t_or_uv_sampling.block(i * samples, 1, samples, 1) = (double)i / (samples - 1) * Eigen::VectorXd::Ones(samples);
        }
        point_sampling.setZero(samples * samples, 3);
        for (int i = 0; i < t_or_uv_sampling.rows(); ++i)
        {
            point_sampling.row(i) = patch.evaluate(t_or_uv_sampling(i, 0), t_or_uv_sampling(i, 1));
        }
 
        {
            Eigen::MatrixXi F(2 * ((samples - 1) * (samples - 1)), 3);
            int f = 0;
            for (int i = 0; i < samples - 1; ++i)
                for (int j = 0; j < samples - 1; ++j)
                {
                    Eigen::MatrixXi F_local(2, 3);
                    F_local << (i * samples + j), ((i + 1) * samples + j), (i * samples + j + 1),
                        (i * samples + j + 1), ((i + 1) * samples + j), ((i + 1) * samples + j + 1);
                    F.block(f, 0, 2, 3) = F_local;
                    f += 2;
                }
            io::OBJWriter::write(fmt::format("spline_target_{:d}.obj", rand() % 100), point_sampling, F);
        }
 
        interpolation_fn = std::make_unique<LazyCubicInterpolator>(dim, delta_);
    }
 
    void SDFTargetForm::compute_distance(const Eigen::MatrixXd &point, double &distance) const
    {
        distance = DBL_MAX;
        Eigen::MatrixXd p = point.transpose();
 
        if (dim == 2)
            for (int i = 0; i < t_or_uv_sampling.size() - 1; ++i)
            {
                const double l = (point_sampling.row(i + 1) - point_sampling.row(i)).squaredNorm();
                double distance_to_perpendicular = ((p - point_sampling.row(i)) * (point_sampling.row(i + 1) - point_sampling.row(i)).transpose())(0) / l;
                const double t = std::max(0., std::min(1., distance_to_perpendicular));
                const auto project = point_sampling.row(i) * (1 - t) + point_sampling.row(i + 1) * t;
                const double project_distance = (p - project).norm();
                if (project_distance < distance)
                    distance = project_distance;
            }
        else if (dim == 3)
        {
            for (int i = 0; i < samples - 1; ++i)
                for (int j = 0; j < samples - 1; ++j)
                {
                    int loc = samples * i + j;
                    const double l1 = (point_sampling.row(loc + 1) - point_sampling.row(loc)).squaredNorm();
                    double distance_to_perpendicular = ((p - point_sampling.row(loc)) * (point_sampling.row(loc + 1) - point_sampling.row(loc)).transpose())(0) / l1;
                    const double u = std::max(0., std::min(1., distance_to_perpendicular));
 
                    const double l2 = (point_sampling.row(loc + samples) - point_sampling.row(loc)).squaredNorm();
                    distance_to_perpendicular = ((p - point_sampling.row(loc)) * (point_sampling.row(loc + samples) - point_sampling.row(loc)).transpose())(0) / l2;
                    const double v = std::max(0., std::min(1., distance_to_perpendicular));
 
                    Eigen::MatrixXd project = point_sampling.row(loc) * (1 - u) + point_sampling.row(loc + 1) * u;
                    project += v * (point_sampling.row(loc + samples) - point_sampling.row(loc));
                    const double project_distance = (p - project).norm();
                    if (project_distance < distance)
                        distance = project_distance;
                }
        }
    }
 
    IntegrableFunctional SDFTargetForm::get_integral_functional() const
    {
        IntegrableFunctional j;
        auto j_func = [this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val.setZero(u.rows(), 1);
 
            for (int q = 0; q < u.rows(); q++)
            {
                double distance;
                Eigen::MatrixXd unused_grad;
                interpolation_fn->evaluate(u.row(q) + pts.row(q), distance, unused_grad);
                val(q) = pow(distance, 2);
            }
        };
 
        auto djdu_func = [this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val.setZero(u.rows(), u.cols());
 
            for (int q = 0; q < u.rows(); q++)
            {
                double distance;
                Eigen::MatrixXd grad;
                interpolation_fn->evaluate(u.row(q) + pts.row(q), distance, grad);
                val.row(q) = 2 * distance * grad.transpose();
            }
        };
 
        j.set_j(j_func);
        j.set_dj_du(djdu_func);
        j.set_dj_dx(djdu_func);
 
        return j;
    }
 
    void MeshTargetForm::set_surface_mesh_target(const Eigen::MatrixXd &V, const Eigen::MatrixXi &F, const double delta)
    {
        dim = V.cols();
        delta_ = delta;
        if ((dim != 3) || (varform_->get_mesh().dimension() != 3))
            log_and_throw_error("MeshTargetForm is only available for 3d scenes.");
 
        tree_.init(V, F);
        V_ = V;
        F_ = F;
 
        interpolation_fn = std::make_unique<LazyCubicInterpolator>(dim, delta_);
    }
 
    void MeshTargetForm::solution_changed_step(const int time_step, const Eigen::VectorXd &x)
    {
        const auto &bases = varform_->primary_space().basis_list();
        const auto &gbases = varform_->primary_space().geometry_basis_list();
        const int actual_dim = varform_->get_problem().is_scalar() ? 1 : dim;
 
        auto storage = utils::create_thread_storage(LocalThreadScalarStorage());
        utils::maybe_parallel_for(varform_->boundary_state().total_local_boundary.size(), [&](int start, int end, int thread_id) {
            LocalThreadScalarStorage &local_storage = utils::get_local_thread_storage(storage, thread_id);
 
            Eigen::MatrixXd uv, samples, gtmp;
            Eigen::MatrixXd points, normal;
            Eigen::VectorXd weights;
 
            Eigen::MatrixXd u, grad_u;
 
            for (int lb_id = start; lb_id < end; ++lb_id)
            {
                const auto &lb = varform_->boundary_state().total_local_boundary[lb_id];
                const int e = lb.element_id();
 
                for (int i = 0; i < lb.size(); i++)
                {
                    const int global_primitive_id = lb.global_primitive_id(i);
                    if (ids_.size() != 0 && ids_.find(varform_->get_mesh().get_boundary_id(global_primitive_id)) == ids_.end())
                        continue;
 
                    utils::BoundarySampler::boundary_quadrature(lb, varform_->n_boundary_samples(), varform_->get_mesh(), i, false, uv, points, normal, weights);
 
                    assembler::ElementAssemblyValues &vals = local_storage.vals;
                    vals.compute(e, varform_->get_mesh().is_volume(), points, bases[e], gbases[e]);
                    io::Evaluator::interpolate_at_local_vals(e, dim, actual_dim, vals, diff_cache_->u(time_step), u, grad_u);
 
                    normal = normal * vals.jac_it[0]; // assuming linear geometry
 
                    for (int q = 0; q < u.rows(); q++)
                        interpolation_fn->cache_grid([this](const Eigen::MatrixXd &point, double &distance) {
                            int idx;
                            Eigen::Matrix<double, 1, 3> closest;
                            distance = pow(tree_.squared_distance(V_, F_, point.col(0), idx, closest), 0.5);
                        },
                                                     vals.val.row(q) + u.row(q));
                }
            }
        });
    }
 
    IntegrableFunctional MeshTargetForm::get_integral_functional() const
    {
        IntegrableFunctional j;
        auto j_func = [this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val.setZero(u.rows(), 1);
 
            for (int q = 0; q < u.rows(); q++)
            {
                double distance;
                Eigen::MatrixXd unused_grad;
                interpolation_fn->evaluate(u.row(q) + pts.row(q), distance, unused_grad);
                val(q) = pow(distance, 2);
            }
        };
 
        auto djdu_func = [this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val.setZero(u.rows(), u.cols());
 
            for (int q = 0; q < u.rows(); q++)
            {
                double distance;
                Eigen::MatrixXd grad;
                interpolation_fn->evaluate(u.row(q) + pts.row(q), distance, grad);
                val.row(q) = 2 * distance * grad.transpose();
            }
        };
 
        j.set_j(j_func);
        j.set_dj_du(djdu_func);
        j.set_dj_dx(djdu_func);
 
        return j;
    }
 
    NodeTargetForm::NodeTargetForm(std::shared_ptr<const varform::DifferentiableVarForm> varform, std::shared_ptr<const DiffCache> diff_cache, const VariableToSimulationGroup &variable_to_simulations, const json &args, const io::ResourceIO &resources)
        : StaticForm(variable_to_simulations), varform_(std::move(varform)), diff_cache_(std::move(diff_cache))
    {
        const int dim = varform_->get_mesh().dimension();
 
        // N x (dim * 2), each row is [rest_x, rest_y, rest_z, deform_x, deform_y, deform_z]
        const Eigen::MatrixXd data = resources.read_matrix(args["target_data_path"].get<std::string>());
 
        // markers to nodes
        target_vertex_positions.setZero(data.rows(), dim);
        active_nodes.reserve(data.rows());
        for (int s = 0; s < data.rows(); s++)
        {
            target_vertex_positions.row(s) = data.block(s, dim, 1, dim);
 
            const RowVectorNd node = data.block(s, 0, 1, dim);
            bool not_found = true;
            double min_dist = std::numeric_limits<double>::max();
            for (int v = 0; v < varform_->primary_space().mesh_nodes->n_nodes(); v++)
            {
                min_dist = std::min(min_dist, (varform_->primary_space().mesh_nodes->node_position(v) - node).norm());
                if ((varform_->primary_space().mesh_nodes->node_position(v) - node).norm() < args["tolerance"])
                {
                    active_nodes.push_back(v);
                    not_found = false;
                    break;
                }
            }
            if (not_found)
                log_and_throw_adjoint_error("Failed to find corresponding node for {}! Minimum distance {}", node, min_dist);
        }
    }
 
    NodeTargetForm::NodeTargetForm(std::shared_ptr<const varform::DifferentiableVarForm> varform, std::shared_ptr<const DiffCache> diff_cache, const VariableToSimulationGroup &variable_to_simulations, const std::vector<int> &active_nodes_, const Eigen::MatrixXd &target_vertex_positions_)
        : StaticForm(variable_to_simulations), varform_(std::move(varform)), diff_cache_(std::move(diff_cache)), target_vertex_positions(target_vertex_positions_), active_nodes(active_nodes_)
    {
        // log_and_throw_adjoint_error("[{}] Constructor not implemented!", name());
    }
 
    Eigen::VectorXd NodeTargetForm::compute_adjoint_rhs_step(const int time_step, const Eigen::VectorXd &x, const varform::DifferentiableVarForm &varform, const DiffCache &diff_cache) const
    {
        Eigen::VectorXd rhs;
        rhs.setZero(diff_cache.u(0).size());
 
        const int dim = varform_->get_mesh().dimension();
 
        if (&varform == varform_.get())
        {
            int i = 0;
            const Eigen::VectorXd disp = diff_cache_->u(time_step);
            for (int v : active_nodes)
            {
                const RowVectorNd cur_pos = varform_->primary_space().mesh_nodes->node_position(v) + disp.segment(v * dim, dim).transpose();
 
                rhs.segment(v * dim, dim) = 2 * (cur_pos - target_vertex_positions.row(i++));
            }
        }
 
        return rhs * weight();
    }
 
    double NodeTargetForm::value_unweighted_step(const int time_step, const Eigen::VectorXd &x) const
    {
        const int dim = varform_->get_mesh().dimension();
        double val = 0;
        int i = 0;
        const Eigen::VectorXd disp = diff_cache_->u(time_step);
        for (int v : active_nodes)
        {
            const RowVectorNd cur_pos = varform_->primary_space().mesh_nodes->node_position(v) + disp.segment(v * dim, dim).transpose();
            val += (cur_pos - target_vertex_positions.row(i++)).squaredNorm();
        }
        return val;
    }
 
    void NodeTargetForm::compute_partial_gradient_step(const int time_step, const Eigen::VectorXd &x, Eigen::VectorXd &gradv) const
    {
        gradv.setZero(x.size());
        gradv = weight() * variable_to_simulations_.apply_parametrization_jacobian(ParameterType::Shape, *varform_, x, [this]() {
            log_and_throw_adjoint_error("[{}] Doesn't support derivatives wrt. shape!", name());
            return Eigen::VectorXd::Zero(0).eval();
        });
    }
 
    BarycenterTargetForm::BarycenterTargetForm(const VariableToSimulationGroup &variable_to_simulations, const json &args, const std::shared_ptr<varform::DifferentiableVarForm> &varform1, std::shared_ptr<const DiffCache> diff_cache1, const std::shared_ptr<varform::DifferentiableVarForm> &varform2, std::shared_ptr<const DiffCache> diff_cache2)
        : StaticForm(variable_to_simulations)
    {
        dim = varform1->get_mesh().dimension();
        json tmp_args = args;
        for (int d = 0; d < dim; d++)
        {
            tmp_args["dim"] = d;
            center1.push_back(std::make_unique<PositionForm>(variable_to_simulations, varform1, diff_cache1, tmp_args));
            center2.push_back(std::make_unique<PositionForm>(variable_to_simulations, varform2, diff_cache2, tmp_args));
        }
    }
 
    Eigen::VectorXd BarycenterTargetForm::compute_adjoint_rhs_step(const int time_step, const Eigen::VectorXd &x, const varform::DifferentiableVarForm &varform, const DiffCache &diff_cache) const
    {
        Eigen::VectorXd term;
        term.setZero(varform.primary_space().ndof());
        for (int d = 0; d < dim; d++)
        {
            double value = center1[d]->value_unweighted_step(time_step, x) - center2[d]->value_unweighted_step(time_step, x);
            term += (2 * value) * (center1[d]->compute_adjoint_rhs_step(time_step, x, varform, diff_cache) - center2[d]->compute_adjoint_rhs_step(time_step, x, varform, diff_cache));
        }
        return term * weight();
    }
    void BarycenterTargetForm::compute_partial_gradient_step(const int time_step, const Eigen::VectorXd &x, Eigen::VectorXd &gradv) const
    {
        gradv.setZero(x.size());
        Eigen::VectorXd tmp1, tmp2;
        for (int d = 0; d < dim; d++)
        {
            double value = center1[d]->value_unweighted_step(time_step, x) - center2[d]->value_unweighted_step(time_step, x);
            center1[d]->compute_partial_gradient_step(time_step, x, tmp1);
            center2[d]->compute_partial_gradient_step(time_step, x, tmp2);
            gradv += (2 * value) * (tmp1 - tmp2);
        }
        gradv *= weight();
    }
    double BarycenterTargetForm::value_unweighted_step(const int time_step, const Eigen::VectorXd &x) const
    {
        double dist = 0;
        for (int d = 0; d < dim; d++)
            dist += std::pow(center1[d]->value_unweighted_step(time_step, x) - center2[d]->value_unweighted_step(time_step, x), 2);
 
        return dist;
    }
 
    MinTargetDistForm::MinTargetDistForm(const VariableToSimulationGroup &variable_to_simulations, const std::vector<int> &steps, const Eigen::VectorXd &target, const json &args, const std::shared_ptr<varform::DifferentiableVarForm> &varform, std::shared_ptr<const DiffCache> diff_cache)
        : AdjointForm(variable_to_simulations), steps_(steps), target_(target)
    {
        dim = varform->get_mesh().dimension();
        json tmp_args = args;
        for (int d = 0; d < dim; d++)
        {
            tmp_args["dim"] = d;
            objs.push_back(std::make_unique<PositionForm>(variable_to_simulations, varform, diff_cache, tmp_args));
        }
        objs.push_back(std::make_unique<VolumeForm>(variable_to_simulations, varform, diff_cache, args));
    }
    Eigen::MatrixXd MinTargetDistForm::compute_adjoint_rhs(const Eigen::VectorXd &x, const varform::DifferentiableVarForm &varform, const DiffCache &diff_cache) const
    {
        Eigen::VectorXd values(steps_.size());
        std::vector<Eigen::MatrixXd> grads(steps_.size(), Eigen::MatrixXd::Zero(varform.primary_space().ndof(), objs.size()));
        Eigen::MatrixXd g2(steps_.size(), objs.size());
        int i = 0;
        for (int s : steps_)
        {
            Eigen::VectorXd input(objs.size());
            Eigen::VectorXd tmp;
            for (int d = 0; d < objs.size(); d++)
            {
                input(d) = objs[d]->value_unweighted_step(s, x);
                grads[i].col(d) = objs[d]->compute_adjoint_rhs_step(s, x, varform, diff_cache);
            }
            values[i] = eval2(input);
            g2.row(i++) = eval2_grad(input);
        }
 
        Eigen::VectorXd g1 = eval1_grad(values);
        Eigen::MatrixXd terms = Eigen::MatrixXd::Zero(varform.primary_space().ndof(), diff_cache.size());
        i = 0;
        for (int s : steps_)
        {
            terms.col(s) += g1(i) * grads[i] * g2.row(i).transpose();
            i++;
        }
 
        return terms * weight();
    }
    void MinTargetDistForm::compute_partial_gradient(const Eigen::VectorXd &x, Eigen::VectorXd &gradv) const
    {
        Eigen::VectorXd values(steps_.size());
        std::vector<Eigen::MatrixXd> grads(steps_.size(), Eigen::MatrixXd::Zero(x.size(), objs.size()));
        Eigen::MatrixXd g2(steps_.size(), objs.size());
        int i = 0;
        for (int s : steps_)
        {
            Eigen::VectorXd input(objs.size());
            Eigen::VectorXd tmp;
            for (int d = 0; d < objs.size(); d++)
            {
                input(d) = objs[d]->value_unweighted_step(s, x);
                objs[d]->compute_partial_gradient_step(s, x, tmp);
                grads[i].col(d) = tmp;
            }
            values[i] = eval2(input);
            g2.row(i++) = eval2_grad(input);
        }
 
        Eigen::VectorXd g1 = eval1_grad(values);
        gradv.setZero(x.size());
        i = 0;
        for (int s : steps_)
        {
            gradv += g1(i) * grads[i] * g2.row(i).transpose();
            i++;
        }
        gradv *= weight();
    }
    double MinTargetDistForm::value_unweighted(const Eigen::VectorXd &x) const
    {
        Eigen::VectorXd values(steps_.size());
        int i = 0;
        for (int s : steps_)
        {
            Eigen::VectorXd input(objs.size());
            for (int d = 0; d < objs.size(); d++)
                input(d) = objs[d]->value_unweighted_step(s, x);
            values[i++] = eval2(input);
        }
 
        return eval1(values);
    }
} // namespace polyfem::solver